diff --git "a/EtAyT4oBgHgl3EQfSfdv/content/tmp_files/load_file.txt" "b/EtAyT4oBgHgl3EQfSfdv/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/EtAyT4oBgHgl3EQfSfdv/content/tmp_files/load_file.txt" @@ -0,0 +1,561 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf,len=560 +page_content='Mechanical feedback linearization of single-input mechanical control systems Marcin Nowicki1 and Witold Respondek2,3 1Poznan University of Technology, Institute of Automatic Control and Robotics, Piotrowo 3a, 61-138 Pozna´n, Poland 2Lodz University of Technology, Institute of Automatic Control, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Stefanowskiego 18, 90-537 Lodz, Poland 3INSA de Rouen Normandie, Laboratoire de Math´ematiques, 76801 Saint-Etienne-du-Rouvray, France January 3, 2023 Abstract We present a new type of feedback linearization that is tailored for me- chanical control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We call it a mechanical feedback linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Its basic feature is preservation of the mechanical structure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For mechanical systems with a scalar control, we formulate necessary and sufficient conditions that are verifiable using differentiations and algebraic operations only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We illustrate our results with several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 1 Introduction An N-dimensional control-affine system with a scalar control ˙z = F(z) + G(z)u, (Σ) where z ∈ Z, an open subset of RN, and u ∈ R, is said to be (locally) feedback linearizable (F-linearizable) if there exist a (local) diffeomorphism Φ : Z → RN and an invertible feedback of the form u = α(z) + β(z)˜u such that the control system (Σ), in the new coordinates ˜z = Φ(z) and with the new control ˜u, is a controllable linear system of the form ˙˜z = A˜z + b˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A geometric solution to the problem of feedback linearization (inspired by [1], and developed independently in [2] and [3]) provides powerful techniques for designing a closed-loop control system that have been used in numerous engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' From a theoretical point of view, that result identifies a class of nonlinear systems that can be considered as linear ones in a well-chosen coordinates and with respect to a well-modified control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='00087v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='OC] 31 Dec 2022 In this paper, we state and study the following fundamental question: if a nonlinear control system (Σ) is mechanical and feedback linearizable, are those two structures compatible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' That is, can we feedback linearize the system pre- serving its mechanical structure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For mechanical control systems, it is natural to consider mechanical feedback equivalence (in particular, to a linear form) under mechanical transformations (coordinates changes and feedback) that pre- serve the mechanical structure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In our recent paper [4], we showed that even in the simplest underactuated case of 2 degrees of freedom, the struc- tures (linear and mechanical) may not conform trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In the present paper, we treat the single-input case in its full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' There are several motivations for preserving the mechanical structure when feedback linearizing the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' First, our formulation of the problem of me- chanical linearization preserves configurations and velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We reckon that it is essential that new configurations (of the linearized system) are functions of the original configurations only, as well as new velocities are true physical velocities (in contrast to pseudo-velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore, we do not lose the physical inter- pretation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This could be useful, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for mechanical systems with constraints on configurations, which are transformed into linear constraints on configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Second, the configuration trajectories are preserved too, which could be useful in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' the motion planning problem (the most natural way to state the problem for mechanical systems is to follow configuration trajectories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Third, it is worth mentioning that mechanical feedback linearizability guaran- tees the linearizing outputs to be functions of configurations only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This may be of constructional importance because one needs only configuration sensors, not those of velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The next argument is the fact that the resultant linear mechanical system allows us to employ dedicated techniques for mechanical sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' An example of such technique is the natural frequency method of tuning a linear feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Finally, when applying mechanical feedback linearization, the physical interpretation of the external action (force, torque, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=') is preserved but is lost for general feedback linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This work is a mechanical counterpart of the classical results on feedback linearization of control systems [1], [2], [3], see also monographs [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Our intention is to formulate conditions for mechanical linearization (shortly, MF- linearization) in a possibly similar manner (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' using involutivity of certain distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For a geometric approach to mechanical control systems see [5], [8], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For mathematical preliminaries concerning the Lie derivative, the Lie bracket, distributions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=', see [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For linearization of mechanical control systems along controlled trajectories see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For mechanical state-space linearization of mechanical control systems see [12] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Compare also [14], for a pioneering work on (partial) feedback linearization of mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Although the state-space of mechanical control system is the tangent bundle TQ of the configuration space Q, we formulate our conditions using objects on Q only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The key here is a geometric approach to mechanical systems [5] and considering the Euler-Lagrange equations as the geodesic equation under an influence of external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 2 The outline of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In Section 2, we state the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In Section 3, we develop further the problem of mechanical feedback linearization and formulate the main result, separately, for mechanical systems with n ≥ 3 in Theorem 1, and with n = 2 in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In Section 4, we provide an application of our results to MF-linearization of several mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Section 6 contains technical results used in proofs that could be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='1 Notation Throughout the Einstein summation convention is assumed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' any expression containing a repeated index (upper and lower) implies the summation over that index up to n, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ωiXi = �n i=1 ωiXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' AT transpose of a matrix (of a vector) A, In n × n identity matrix, Q configuration manifold, X(Q) the set of smooth vector fields on a manifold Q, TxQ tangent space at x ∈ Q, TQ = � x∈Q TxQ tangent bundle of Q, x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' xn) a local coordinate system on Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' φ a diffeomorphism of Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and Φ a diffeomorphism of TQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Dφ = ∂φ ∂x the Jacobian matrix of a diffeomorphism φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂˜xi ∂xj := ∂φi ∂xj the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' j)-element of the Jacobian matrix Dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂xj ∂˜xi the (j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' i)-element of the inverse of the Jacobian matrix Dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' LXα Lie derivative of a function α defined as LXα = ∂α ∂xi Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' [X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Y ] = ∂Y ∂x X − ∂X ∂x Y = adXY Lie bracket of vector fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂ ∂xi the i-th unity vector field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and dxi the i-th unity covector field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' in a co- ordinate system x = (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , xn), Ei = span � adj eg, 0 ≤ j ≤ i � distribution on Q spanned by adj eg, ∇ covariant derivative, and ∇2 second covariant derivative, Γi jk Christoffel symbols of the second kind of ∇, 2 Problem statement Consider an n-dimensional configuration space Q (an open subset of Rn or, in general, an n-dimensional manifold) equipped with a symmetric affine connec- tion ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The operator of the affine connection ∇ allows to define intrinsically the acceleration as the covariant derivative ∇ ˙x(t) ˙x(t), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' [5,8,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The covari- ant derivative ∇ : X(Q) × X(Q) → X(Q) of an arbitrary vector field Y = Y i ∂ ∂xi with respect to X = Xi ∂ ∂xi in coordinates reads ∇XY = �∂Y i ∂xj Xj + Γi jkXjY k � ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (1) 3 A mechanical control system (MS) is a 4-tuple (Q, ∇, g, e), where g and e are, respectively, controlled and uncontrolled vector fields on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A curve x(t) : I → Q, I ⊂ R, is a trajectory of (MS) if it satisfies the following equation ∇ ˙x(t) ˙x(t) = e (x(t)) + g (x(t)) u, (2) which can be viewed as an equation that balances accelerations of the system, where the left-hand side represents geometric accelerations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' accelerations caused by the geometry of the system) and the right-hand side represents ac- celerations caused by external actions on the system (controlled or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Notice that (2) is a second-order differential equation on Q (indeed, using (1) we con- clude that ∇ ˙x ˙x depends on ¨x, see [5] for details) and can be rewritten as a system of first-order differential equations on TQ, which we also call a mechan- ical control system (MS): ˙xi = yi ˙yi = −Γi jk(x)yjyk + ei(x) + gi(x)u, (MS) for 1 ≤ i ≤ n, where (x, y) = � x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , xn, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , yn� are local coordinates on the tangent bundle TQ of the configuration manifold Q, and Γi jk(x) are Christoffel symbols of the affine connection ∇ that correspond to the Cori- olis and centrifugal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The vector fields e(x) = (e1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , en(x))T and g(x) = (g1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , gn(x))T correspond to, respectively, uncontrolled and con- trolled actions on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Throughout all objects are assumed to be smooth and the word smooth means C∞-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Our obvious inspirations are Lagrangian mechanical control systems without dissipative forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For the correspondence between (MS) and the Lagrangian equations of dynamics see [5], [8], [9] and our recent papers [13], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' However, we will consider throughout a more general class of mechanical control systems allowing for any symmetric (not necessarily a metric) connection and any (not necessarily potential) vector field e(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Consider the group of mechanical feedback transformations GMF generated by the following transformations: (i) changes of coordinates in TQ given by Φ : TQ → T ˜Q (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' y) �→ (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ˜y) = Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' y) = � φ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂φ ∂x(x)y � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (3) called a mechanical diffeomorphism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where φ : Q → ˜Q is a diffeomorphism and ∂φ ∂x its Jacobian matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (ii) mechanical feedback transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' denoted (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' of the form u = γjk(x)yjyk + α(x) + β(x)˜u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (4) where γjk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' β are smooth functions on Q satisfying γjk = γkj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' β(·) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The matrix γ = (γjk) represents a (0, 2)−tensor field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 4 Even if the diffeomorphism φ is possibly local on Q, the action of ∂φ ∂x(x) is always global on fibers TxQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The system (MS) is MF-linearizable if there exist mechanical feedback transformations (Φ, α, β, γ) ∈ GMF bringing (MS) into a linear con- trollable mechanical system of the form ˙˜xi = ˜yi ˙˜yi = Ei j ˜xj + bi˜u, (LMS) where (˜x, ˜y) are coordinates on TRn = Rn × Rn, the matrix E = (Ei j) is an n × n real-valued matrix, the vector field b = bi ∂ ∂˜xi is constant, and the pair (E, b) is controllable (see [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Represent (MS) as ˙z = F(z) + G(z)u, where z = (x, y) ∈ TQ, F = yi ∂ ∂xi + � −Γi jk(x)yjyk + ei(x) � ∂ ∂yi , and G = gi(x) ∂ ∂yi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The problem that we formulate and solve in the paper is whether (MS) is MF-linearizable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' That is, do there exist Φ = (˜x, ˜y) = (φ, ∂φ ∂xy) and (α, β, γ) such that ∂Φ ∂z (z) � F + G(yT γy + α) � (z) = � ˜y E˜x � , ∂Φ ∂z (z) (Gβ) (z) = �0 b � ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Note that MF-linearizability is stronger than the classical feedback lineariz- ability since, for the latter, Φ : TQ → R2n can be any diffeomorphism (need not be of mechanical form (3)) and yT γ(x)y +α(x) can be replaced by any function α(x, y) on TQ and β(x) by any invertible function β(x, y) on TQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' If we neglect the mechanical structure of ˙z = F(z) + G(z)u, and consider it as a general control system, we can ask if the system is F-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The well-known answer [2,3] asserts that, locally, this is the case if and only if the distributions Di = span � adj F G, 0 ≤ j ≤ i � are involutive and of constant rank for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=', 2n − 1 and D2n−1 = TQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The natural question arises whether, for F-linearizable (MS), the general feedback transformations (Φ(z), α(z), β(z)) are mechanical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' of the form (3) and (4)) or whether they can be replaced by mechanical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Example 1: Consider the mechanical system ˙x1 = y1 ˙x2 = y2 ˙y1 = −x2(y1)2 + x2 ˙y2 = u, (5) on R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This system is locally F-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Indeed, the local diffeomorphism ˜z = Φ(z), where z = (x1, x2, y1, y2), ˜z = (˜x1, ˜x2, ˜y1, ˜y2), given by ˜x1 = x1 ˜x2 = x2 − x2(y1)2 ˜y1 = y1 ˜y2 = � (y1)2 − 1 � � 2(x2)2y1 − y2� , 5 together with the feedback u = 2(x2)3 +6(x2 −(x2)2)(y1)2 + ˜u (y1)2−1, render the original system linear and controllable ˙˜x1 = ˜y1 ˙˜x2 = ˜y2 ˙˜y1 = ˜x2 ˙˜y2 = ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore, the system is F-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Note, however, that neither the change of coordinates nor the feedback is mechanical (˜x2 depends on velocities, and the function β depends on velocities as well) so the mechanical structure is not preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Our question is whether this system can be linearized by other transformations that preserve the mechanical structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' can it be MF- linearized?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The group of mechanical transformations GMF = {(Φ, α, β, γ)} preserves trajectories, that is, maps the trajectories of (MS) into those of its MF-equivalent system ( � MS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Indeed, if z (t, z0, u(t)) is a trajectory of (MS) (passing through z0 = (x0, y0) and corresponding to a control u(t)), then ˜z (t, ˜z0, ˜u(t)) = Φ (z (t, z0, u(t))) is a trajectory of ( � MS) passing through ˜z0 = Φ(z0) = (φ(x0), ∂φ ∂x(x0)y0) and corresponding to ˜u(t), where u(t) = y(t)T γ (x(t)) y(t) + α (x(t)) + β (x(t)) ˜u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Moreover, via φ : Q → ˜Q, it establishes a correspondence between configuration trajectories in Q and ˜Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ˜x (t, ˜z0, ˜u(t)) = φ (x(t, z0, u(t))), making the fol- lowing diagram commutative (notice, however, that π (z(t, z0, u)) = x(t, z0, u) depends on z0 = (x0, y0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' an initial configuration x0 and initial velocity y0): z(t, z0, u) ˜z(t, ˜z0, ˜u) x(t, z0, u) ˜x(t, ˜z0, ˜u) (Φ,α,β,γ) π π (φ,α,β,γ) where π : TQ → Q, π(z) = π(x, y) = x, is the canonical projection which assigns to the pair (x, y) the point x at which the velocity y is attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 3 Mechanical feedback linearization Our main result uses two basic ingredients: the covariant derivative of the con- nection ∇, see (1), and the involutivity of suitable distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will also need the second covariant derivative of a vector field Z in the directions (X, Y ), which is a mapping ∇2 : X(Q) × X(Q) × X(Q) → X(Q) ∇2 X,Y Z = ∇X∇Y Z − ∇∇XY Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (6) For properties of the second covariant derivative see Lemma 1 in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In order to formulate the result, we associate with (MS) the following se- quence of nested distributions E0 ⊂ E1 ⊂ E2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ⊂ Ei ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ⊂ TQ, where E0 = span {g} , Ei = span � adj eg, 0 ≤ j ≤ i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 6 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To analyze the behavior of the distributions Ei under mechanical feedback transformations (α, β, γ) notice, first, that Ei are invariant under γ since γ does not act on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' If the distributions Ei are involutive, then they are invariant under feedback transformations of the form (α, β, 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for γ = 0 they remain unchanged if we replaced e and g by, respectively, e + gα and βg, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, we formulate our main result for MF-linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' First, we state a theorem for (MS) with n ≥ 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The remaining case of n = 2 degrees of freedom is treated in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For an explanation of that distinction, see the comment before Theorem 2 and Remark 3 for a comparison of both results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By a local MF-linearization around x0 ∈ Q we mean that it holds on � x∈O TxQ, where O is a neighborhood of x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' recall that all transformations are global on tangent spaces TxQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Assume n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A mechanical control system (MS) is, locally around x0, MF-linearizable to a controllable (LMS) if and only if (MF1) rank En−1 = n, (MF2) Ei is involutive and of constant rank, for 0 ≤ i ≤ n − 2, (MF3) ∇adieg g ∈ E0 for 0 ≤ i ≤ n − 1, (MF4) ∇2 adkeg,adj eg e ∈ E1 for 0 ≤ k, j ≤ n − 1, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Notice that (MF1)-(MF2) are the classical conditions (see [2,3,6, 7]) that assure F-linearization of the system ˙x = e(x)+g(x)u on Q via ˜x = φ(x) and u = α(x)+β(x)˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The remaining two, (MF3)-(MF4), can be interpreted as compatibility conditions that guarantee vanishing the Christoffel symbols Γi jk in the linearizing coordinates ˜x = φ(x), except for those that can be compensated by feedback u = γjk(x)yjyk + ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In the proof we will use two Lemmata 1 and 2, given in Appendix, that are of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For (LMS), we have Γi jk = 0, e = Ex and g = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows that adi eg = (−1)iEib and therefore, using the definitions of ∇, given by (1), and of ∇2, given by (6), we calculate ∇adiegadj eg = 0, ∇2 adkeg,adj ege = 0, (7) which implies that (MF1)-(MF4) hold for (LMS) (in particular, (MF1) holds because (LMS) is assumed controllable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To prove necessity of (MF1)-(MF4), we will show that they are MF-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' All conditions (MF1)-(MF4) are expressed in a geometrical way, therefore they are invariant under diffeomor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The conditions (MF1) and (MF2) are mechanical feedback invariant, see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It remains to show that (MF3) and (MF4) are invariant under the 7 mechanical feedback u = γjk(x)yjyk+α(x)+β(x)˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For the closed-loop system, denoted by ”∼”, the Christoffel symbols ˜Γi jk of ˜∇, ˜e, and ˜g are, respectively, given by ˜Γi jk = Γi jk − giγjk, ˜e = e + gα, ˜g = gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (8) For any X, Y ∈ X(Q), we have ˜∇XY = ∇XY − γ(X, Y )g = ∇XY mod E0, where γ(X, Y ) = γjkXjY k ∈ C∞(Q), therefore ˜∇adi ˜e˜g˜g = ∇adi ˜e˜g˜g − γ(adi ˜e˜g, ˜g)g = ∇adi ˜e˜g˜g mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By ∇X˜g = ∇X (gβ) = ∇Xg + (LXβ) g, it follows that instead of calculating ∇adi ˜e˜g˜g it is enough to calculate ∇adi ˜e˜gg, since the second term (LXβ) g ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For i=0, we have ∇˜gg = ∇(gβ)g = β∇gg ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It is easy to show that for any 1 ≤ j ≤ n − 1, we have adj ˜e˜g = βadj eg + dj−1, (9) where dj−1 ∈ Ej−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Assume ∇adl ˜e˜gg ∈ E0, for 0 ≤ l ≤ i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then, by formula (9), ∇adi ˜e˜gg = β∇adiegg + ∇di−1g ∈ E0, because the first term is in E0 by (MF3) and the second by the induction assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We have thus proved necessity of (MF3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To show necessity of (MF4), using Lemma 1, calculate ˜∇2 X,Y Z = ˜∇X ˜∇Y Z − ˜∇ ˜∇XY Z = ˜∇X (∇Y Z − γ(Y, Z)g) − ˜∇(∇XY −γ(X,Y )g)Z = ∇2 X,Y Z − γ(Y, Z)∇Xg + γ(X, Y )∇gZ mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (10) By the above formula, we get ˜∇2 adk ˜e ˜g,adj ˜e˜g˜e =∇2 adk ˜e ˜g,adj ˜e˜g˜e − γ(adj ˜e˜g, ˜e)∇adk ˜e ˜gg + γ(adk ˜e˜g, adj ˜e˜g)∇g˜e mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The second term, on the right hand side, is in E0 (by (MF3) and its invariance), while the third term is a function multiplying ∇g˜e = ∇g (e + gα) = ∇ge + α∇gg + Lgα g ∈ E1, since for (LMS) we have ∇ge = −adeg = −Eb ∈ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The first term ∇2 adk ˜e ˜g,adj ˜e˜g˜e is, by (9) and Lemma 1(i), a linear combination with smooth coefficients of ∇2 adieg,adleg˜e, with 0 ≤ i ≤ k and 0 ≤ l ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Thus we calculate ∇2 adi eg,adl eg˜e = ∇2 adieg,adlege+∇2 adieg,adleg(gα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The first term vanishes since (7) holds for (LMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We calculate the second term using Lemma 1(iii), and we have ∇2 adi eg,adl eg(gα) = α∇2 adieg,adlegg + Ladi egα∇adlegg + Ladlegα∇adi egg + (∇2 adi eg,adl egα)g ∈ E0 because the first three terms vanish, due to (7), and 8 the last one is in E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Summarizing the above calculations, we conclude that ˜∇2 adk ˜e ˜g,adj ˜e˜g˜e ∈ E1 = ˜E1, which proves necessity of (MF4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will transform the system (MS), satisfying (MF1)-(MF4), into (LMS) in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In the first step, we will normalize the vector fields e and g and show that condition (MF4) implies zeroing some of the Christoffel symbols Γi jk, which exhibit a triangular form in the normalizing coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In the second step, we compensate the remaining Christoffel symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By conditions (MF1)-(MF2), there exists a function h satisfying Ladj egh = 0, for 0 ≤ j ≤ n − 2, and Ladn−1 e gh ̸= 0, and thus (˜x, ˜y) = (φ(x), ∂φ ∂x(x)y) is a local mechanical diffeomorphism, where φ(x) = (Ln−1 e h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , Leh, h)T that can be completed by a feedback transformation (α, β, 0) that map, respectively, βg into ˜g = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , 0)T , e + gα into ˜e = (0, ˜x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , ˜xn−1)T , and Γi jk into ˜Γi jk, see the classical results of feedback linearization [2], [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then, (Φ, α, β, γ) ∈ GMF , where (˜x, ˜y) = Φ(x, y) = � φ(x), ∂φ ∂x(x)y � with φ, α, β just defined and γjk = ˜Γ1 jk(˜x), brings (MS) into (we drop ”tildas” for readability) ˙x1 = y1 ˙xi = yi ˙y1 = u ˙yi = −Γi jkyjyk + xi−1, 2 ≤ i ≤ n, (11) to which Lemma 2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will show that the Christoffel symbols Γi jk of (11) satisfy Γi kj = 0 for 1 ≤ k ≤ n − 1, 1 ≤ j ≤ i ≤ n, Γi nj = � 0 for 1 ≤ j < i ≤ n λ(xn) for 2 ≤ j = i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (12) For system (11), we have adk−1 e g = (−1)k−1 ∂ ∂xk and, in particular, g = ∂ ∂x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Calculate ∇adk−1 e gg = (−1)k−1∇ ∂ ∂xk gi ∂ ∂xi = (−1)k−1∇ ∂ ∂xk ∂ ∂x1 = (−1)k−1Γi k1 ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows that Γi k1 = Γi 1k = 0, for 2 ≤ i ≤ n by (MF3), and for i = 1 by the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Rewrite (MF4) as ∇2 adk−1 e g,adj−1 e ge = 0 mod E1, for 1 ≤ j, k ≤ n, and apply it successively for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , n and for all 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For j = 1, first calculate ∇ge = ∇ ∂ ∂x1 e = ∂ ∂x2 + Γi 1ses ∂ ∂xi = ∂ ∂x2 and then ∇adk−1 e g (∇ge) = (−1)k−1∇ ∂ ∂xk ∂ ∂x2 = (−1)k−1Γi k2 ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' On the other hand, ∇adk−1 e gg = (−1)k−1∇ ∂ ∂xk ∂ ∂x1 = (−1)k−1Γ1 k1 ∂ ∂x1 = 0 and hence ∇∇adk−1 e gge = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Thus, by (6), ∇2 adk−1 e g,ge = ∇adk−1 e g (∇ge) − ∇∇adk−1 e gge = = (−1)k−1Γi k2 ∂ ∂xi = 0 mod E1, 9 implying that Γi k2 = Γi 2k = 0 for any 3 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For j = 2, calculate ∇adege = −∇ ∂ ∂x2 e = − ∂ ∂x3 + Γi 2ses ∂ ∂xi = − ∂ ∂x3 − d where d = d1(x) ∂ ∂x1 + d2(x) ∂ ∂x2 ∈ E1, and then ∇adk−1 e g (∇adege) = (−1)k∇ ∂ ∂xk � ∂ ∂x3 + d � = = (−1)k � Γi k3 + Γi k1d1 + Γi k2d2� ∂ ∂xi = = (−1)kΓi k3 ∂ ∂xi mod E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' On the other hand, ∇adk−1 e gadeg = (−1)k∇ ∂ ∂xk ∂ ∂x2 = (−1)kΓi k2 ∂ ∂xi = = (−1)k � Γ1 k2 ∂ ∂x1 + Γ2 k2 ∂ ∂x2 � and ∇∇adk−1 e gadege = (−1)kΓ2 k2 ∂ ∂x3 mod E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows that, modulo E1, ∇2 adk−1 e g,adege = (−1)k � n � i=4 Γi k3 ∂ ∂xi + (Γ3 k3 − Γ2 k2) ∂ ∂x3 � , and, using (MF4), we conclude Γi k3 = Γi 3k = 0 for any 4 ≤ i ≤ n and Γ3 k3 = Γ2 k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Following the same line (with a more tedious calculation), one can prove the general induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Namely, assuming, for a fixed j, Γj kj = Γj−1 kj−1 Γi ks = Γi sk = 0 s + 1 ≤ i ≤ n, 1 ≤ s ≤ j, (13) one shows by calculating ∇2 adk−1 e g,adj−1 e ge, with the help of (24) of Lemma 2, that Γj+1 kj+1 = Γj kj Γi kj+1 = 0 for j + 2 ≤ i ≤ n and thus, by the induction assumption and symmetry of the Christoffel symbols, Γi ks = Γi sk = 0 s + 1 ≤ i ≤ n, 1 ≤ s ≤ j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (14) It follows that for each 1 ≤ k ≤ n the matrices consisting of Christoffel symbols (Γi kj), for 2 ≤ i, j ≤ n are upper triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By the induction argument, (13) holds for all 2 ≤ j ≤ n and implies, for any 1 ≤ k ≤ n − 1, Γ2 k2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' = Γn−1 kn−1 = Γn kn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 10 since Γn kn = Γn nk = 0 (as n > k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' On the other hand, for k = n, (13) implies Γ2 n2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' = Γn−1 nn−1 = Γn nn = λ(x) for a function λ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore for each 1 ≤ k ≤ n the matrices (Γi kj), for 2 ≤ i, j ≤ n, are strictly upper triangular, and the last one, for k = n, is upper triangular with all diagonal elements equal to each other, which we denote by λ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The matrices read � Γi kj � = � � � � � � � � � 0 Γ2 k3 Γ2 k4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ2 kn−2 Γ2 kn−1 Γ2 kn 0 0 Γ3 k4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ3 kn−2 Γ3 kn−1 Γ3 kn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 Γn−2 kn−1 Γn−2 kn 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 Γn−1 kn 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 0 � � � � � � � � � , for 1 ≤ k ≤ n − 1, and � Γi nj � = � � � � � � � � � λ Γ2 n3 Γ2 n4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ2 nn−2 Γ2 nn−1 Γ2 nn 0 λ Γ3 n4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ3 nn−2 Γ3 nn−1 Γ3 nn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' λ Γn−2 nn−1 Γn−2 nn 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 λ Γn−1 nn 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 λ � � � � � � � � � , and are thus of the desired triangular structure (12) and it remains to prove that λ = λ(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Note that in the above matrices we skip the first row Γ1 kj and the first column Γi k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This is due to the fact that Γ1 kj = 0 (which can always be achieved by a suitable feedback transformation) and Γi k1 = 0 by (MF3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Notice that we have En−2 = span � ∂ ∂x1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂ ∂xn−1 � and thus applying (24) of Lemma 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for j = n and any 1 ≤ k ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' we conclude (set Γn kn+1 = 0) (−1)n+k−2∇2 adk−1 e g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adn−1 e ge = ∇2 ∂ ∂xk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂ ∂xn e = �∂Γn ns ∂xk es + Γn nk+1 + Γn kn+1 − Γn−1 kn + (Γd nsΓn kd − Γd knΓn ds)es � ∂ ∂xn mod En−2 = � ∂λ ∂xk en + Γn nk+1 − Γn−1 kn � ∂ ∂xn mod En−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (15) since,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' due to the triangular structure (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γn ns = 0 except for s = n giving Γn nn = λ and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' the equality Γd nsΓn kd−Γd knΓn ds = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Indeed, in the latter, Γn kd = 0 except d = k = n giving Γn nsΓn nn − Γn nnΓn ns = 0 and Γn ds = 0 except for d = s = n giving Γn nnΓn kn − Γn knΓn nn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 11 For (15) we will apply (MF4) in three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' First, if 1 ≤ k ≤ n − 2, then, modulo En−2, we have � ∂λ ∂xk en + Γn nk+1 − Γn−1 kn � ∂ ∂xn = � ∂λ ∂xk xn−1 � ∂ ∂xn = 0, since all Γn nk+1 = 0 and all Γn−1 kn = 0 by (14) and k ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Second, for k = n − 1, we have modulo En−2, � ∂λ ∂xn−1 en + Γn nn − Γn−1 n−1n � ∂ ∂xn = � ∂λ ∂xn−1 en + λ − λ � ∂ ∂xn = � ∂λ ∂xn−1 xn−1 � ∂ ∂xn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore ∂λ ∂xk = 0, for 1 ≤ k ≤ n − 1, implying that λ is a function of the last variable xn only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' λ = λ(xn), which gives the system in the desired form (12) Third, for k = n, we have modulo En−2, � ∂λ ∂xn en+Γn nn+1−Γn−1 nn � ∂ ∂xn = � ∂λ ∂xn xn−1− Γn−1 nn � ∂ ∂xn = 0, implying that Γn−1 nn = Leλ, since ∂λ(xn) ∂xn xn−1 = Leλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, transform system (11), satisfying (12), via the local mechanical diffeo- morphism Φ : TQ → T ¯Q ¯x = φ(x) ¯y = Dφ(x)y, where φ(x) = � Ln−1 e h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , Leh, h �T , (16) with h(xn) = � xn 0 Λ(s2)ds2, where Λ(s2) = exp �� s2 0 λ(s1)ds1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Denote by ¯Γi jk, ¯e, ¯g the objects of the system expressed in coordinates ¯x = φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Applying feedback ¯u = −¯Γ1 jk¯yj ¯yk + Ln e h + uLgLn−1 e h, the transformed system becomes ˙¯x1 = ¯y1 ˙¯xi = ¯yi ˙¯y1 = ¯u ˙¯yi = −¯Γi jk¯yj ¯yk + ¯xi−1, 2 ≤ i ≤ n, (17) whose vector fields are ¯e = ¯xi−1 ∂ ∂¯xi , where x0 = 0, and ¯g = ∂ ∂¯x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Transformed system (17) is still of the form (11) and at the moment we ignore how Γi jk have been changed into ¯Γi jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Below we will prove that all ¯Γi jk vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To this end, we first calculate explicitly the time-evolution of the pair (¯xn, ¯yn) ˙¯xn = d dth(xn) = Λ(xn) ˙xn = Λ(xn)yn = ¯yn ˙¯yn = d dt (Λ(xn)yn) = Λ(xn)λ(xn) ˙xnyn + Λ(xn) ˙yn = Λ(xn)λ(xn)ynyn + Λ(xn) ˙yn = Λ(xn)λ(xn)ynyn + Λ(xn) � −Γn nn(xn)ynyn + xn−1� = Λ(xn)xn−1 = ¯xn−1, 12 since ¯xn−1 = Leh = Λ(xn)xn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows that ¯Γn jk = 0, for all 1 ≤ k, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For transformed system (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' we rewrite (24) by adding ”bars” as ∇2 adk−1 ¯e ¯g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adj−1 ¯e ¯g¯e = (−1)j+k �∂¯Γi js ∂¯xk ¯es + ¯Γi jk+1 + ¯Γi kj+1 + (¯Γd js¯Γi kd − ¯Γd kj ¯Γi ds)¯es − ¯Γi−1 kj � ∂ ∂¯xi (18) and by (MF4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' we have ∇2 adk−1 ¯e ¯g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adj−1 ¯e ¯g¯e = (−1)j+k¯an kj(¯x) ∂ ∂¯xn = 0 mod En−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where ¯an kj(¯x) = ∂¯Γn js ∂¯xk ¯es + ¯Γn jk+1 + ¯Γn kj+1 + (¯Γd js¯Γn kd − ¯Γd kj ¯Γn ds)¯es − ¯Γn−1 kj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' which implies (since ¯Γn kj = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for 1 ≤ j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' k ≤ n) that ¯an kj(¯x) = ¯Γn−1 kj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now assume ¯Γi kj = 0 for a certain 1 ≤ i ≤ n − 1 and any 1 ≤ j, k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then (18) and (MF4) imply ¯Γi−1 kj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we have proved that all Christoffel symbols of (17) vanish and thus the system is a linear controllable (LMS), since the vector field ¯e = ¯xi−1 ∂ ∂¯xi is linear and ¯g = ∂ ∂¯x1 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The above theorem does not work for systems with 2 degrees of freedom, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for n=2, as that case is too restrictive for involutivity, see Remark 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we state the following theorem for MF-linearization of (MS) with 2 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A mechanical system (MS) with 2 degrees of freedom is, locally around x0, MF-linearizable to a controllable linear (LMS) if and only if it satisfies in a neighborhood of x0 (MF1)’ g and adeg are independent at x0, (MF3)’ ∇g g ∈ E0 and ∇adeg g ∈ E0, (MF5)’ ∇2 g,adeg adeg − ∇2 adeg,g adeg ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' If n = 2, then E0 is of rank 1, thus involutive and (MF2) is trivially satisfied, and so is (MF4) because E1 = TQ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore (MF2)’ and (MF4)’ are absent and replaced by (MF5)’ that guarantees that we can compensate the Christoffel symbols (as do (MF3)-(MF4) for n ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Note that (MF1)’ is equivalent to (MF1) and (MF3)’ is (MF3) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Although Theorem 1 applies to n ≥ 3, the necessity part of its proof remains valid for any n ≥ 2 so it shows necessity of (MF1)’-(MF3)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we need to show necessity of (MF5)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For a controllable (LMS) we have Γi jk = 0, g = b and adeg = −Eb are independent, and ∇adiegadj eg = 0, ∇2 adj eg,adkegadi eg = 0, � adj eg, adk eg � = 0, (19) 13 for 0 ≤ i, j, k ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will use formula (10) to show that (MF5)’ is invariant under mechanical feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Denote ˜∇, ˜e, ˜g, γ as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then we calculate ˜∇2 ˜g,ad˜e˜gad˜e˜g =∇2 ˜g,ad˜e˜gad˜e˜g − γ(ad˜e˜g, ad˜e˜g)∇˜g˜g + γ(˜g, ad˜e˜g)∇˜gad˜e˜g mod E0, ˜∇2 ad˜e˜g,˜gad˜e˜g =∇2 ad˜e˜g,˜gad˜e˜g − γ(g, ad˜e˜g)∇ad˜e˜g˜g + γ(ad˜e˜g, ˜g)∇˜gad˜e˜g mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The second terms of the right hand side of both equations are in E0 due to the feedback invariance of (MF3)’, while the third terms are equal since γ(X, Y ) = γ(Y, X) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we conclude ˜∇2 ˜g,ad˜e˜gad˜e˜g − ˜∇2 ad˜e˜g,˜gad˜e˜g = ∇2 ˜g,ad˜e˜gad˜e˜g − ∇2 ad˜e˜g,˜gad˜e˜g mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Denoting ad˜e˜g = βadeg + d0g (see (9)) and by Lemma 1 (i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' we have ∇2 ˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='ad˜e˜gad˜e˜g = ∇2 βg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='βadeg+d0gad˜e˜g = β2∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adegad˜e˜g + βd0∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g ∇2 ad˜e˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='˜gad˜e˜g = ∇2 βadeg+d0g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='βgad˜e˜g = β2∇2 adeg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g + βd0∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where the last terms on the right are equal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' implying ∇2 ˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='ad˜e˜gad˜e˜g − ∇2 ad˜e˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='˜gad˜e˜g = β2 � ∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adegad˜e˜g − β2∇2 adeg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g � and it remains to prove that ∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adegad˜e˜g − ∇2 adeg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g ∈ E0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' which we show using Lemma 1(iii),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Y stand for either g or adeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Denote ∇Xβ = LXβ and ∇2 X,Y β = LXLY β − L∇XY β (see Lemma 1) and calculate ∇2 X,Y ad˜e˜g = ∇2 X,Y � βadeg + d0g � = β∇2 X,Y adeg + LXβ∇Y adeg + LY β∇Xadeg + � ∇2 X,Y β � adeg + d0∇2 X,Y g + LXd0∇Y g + LY d0∇Xg + � ∇2 X,Y d0� g = � ∇2 X,Y β � adeg mod E0, since all ∇2 X,Y X = 0 and ∇XY = 0 , see (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we have ∇2 g,adegad˜e˜g − ∇2 adeg,gad˜e˜g = � ∇2 g,adegβ − ∇2 adeg,gβ � adeg mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Finally, we calculate ∇2 g,adegβ − ∇2 adeg,gβ = LgLadegβ − L∇gadegβ − � LadegLgβ − L∇adeggβ � = L[g,adeg]β = 0, 14 which shows necessity of (MF5)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By (MF1)’, rank E1 = 2, and E0 = span {g} is of constant rank 1 and thus always involutive, hence the system is, locally around x0 (since g(x0) ̸= 0), MF-equivalent to (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (11)) ˙x1 = y1 ˙x2 = y2 ˙y1 = u ˙y2 = −Γ2 jkyjyk + x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We have g = ∂ ∂x1 , adeg = − ∂ ∂x2 and now we calculate ∇gg = Γ2 11 ∂ ∂x2 ∇adegg = −Γ2 12 ∂ ∂x2 , which by (MF3)’ are in E0 = span � ∂ ∂x1 � , implying Γ2 11 = Γ2 12 = Γ2 21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows ∇gg = ∇adegg = ∇gadeg = 0, and ∇adegadeg = Γ2 22 ∂ ∂x2 and thus ∇2 g,adeg adeg − ∇2 adeg,g adeg = ∇g∇adegadeg − ∇∇gadegadeg − ∇adeg∇gadeg − ∇∇adeggadeg = ∇g∇adegadeg = ∇ ∂ ∂x1 Γ2 22 ∂ ∂x2 = ∂Γ2 22 ∂x1 ∂ ∂x2 implying, by (MF5)’, ∂Γ2 22 ∂x1 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ2 22(x2) = λ(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, we transform the system via the local mechanical diffeomorphism Φ : TQ → T ¯Q (compare to (16)) ¯x = φ(x) ¯y = Dφ(x)y, where φ(x) = (Leh, h)T , with h(x2) = � x2 0 Λ(s2)ds2 and Λ(s2) = exp �� s2 0 λ(s1)ds1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We calculate the evolution of the pair (¯x(t), ¯y(t)) of transformed coordinates, using d dth � x2(t) � = Λ � x2(t) � ˙x2(t) and d dtΛ � x2(t) � = λ � x2(t) � Λ � x2(t) � ˙x2(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' first we get ˙¯x2 = d dth(x2) = Λ(x2)y2 = ¯y2 ˙¯y2 = Λ(x2)λ(x2)y2y2 + Λ(x2) ˙y2 = Λ(x2)λ(x2)y2y2 + Λ(x2) � −λ(x2)y2y2 + x2� = Λ(x2)x1 = ¯x1 and then ˙¯x1 = Λ(x2)y1 + d dtΛ � x2(t) � x1y2 = ¯y1 ˙¯y1 = −¯Γ1 jk¯yj ¯yk + L2 eh + uLgLeh, where we denote by ¯Γ1 jk the Christoffel symbols in the ˙¯y1-equation of the trans- formed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Applying the feedback ¯u = −¯Γ1 jk¯yj ¯yk + L2 eh + uLgLeh, we get a controllable linear mechanical system in the canonical form ˙¯x1 = ¯y1, ˙¯y1 = ¯u, ˙¯x2 = ¯y2, ˙¯y2 = ¯x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 15 4 Examples Example 1 (cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ): For system (5), we have g = ∂ ∂x2 and adeg = − ∂ ∂x1 are in- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We check MF-linearizability using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A simple calculation shows that ∇gg = ∇adegg = 0 ∈ E0, but ∇2 g,adeg adeg−∇2 adeg,g adeg = ∂ ∂x1 /∈ E0, therefore the system is not MF-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Thus (5) is an example of a system that is F-linearizable but not MF- linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For such systems the choice is: either to F-linearize for the price of loosing the mechanical structure or to keep the mechanical structure but to get rid of the linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Example 2: Consider the equation of dynamics of the Inertia Wheel Pen- dulum [18] with constant parameters m0, md, J2: ˙x1 = y1, ˙x2 = y2, ˙y1 = e1 + g1u, ˙y2 = e2 + g2u, e1 = m0 md sin x1, e2 = − m0 md sin x1, g1 = − 1 md , g2 = md+J2 J2md .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will verify whether the conditions of Theorem 2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' First, we calculate adeg = ( m0 m2 d cos x1) ∂ ∂x1 − ( m0 m2 d cos x1) ∂ ∂x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It can be checked that g and adeg are independent for x1 ̸= ± π 2 , which corresponds to the horizontal position of the pendulum, therefore (MF1)’ is satisfied everywhere except for x1 = ± π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Next, we verify condition (MF2)’ by calculating ∇gg = ∇adegg = 0 ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Finally, a direct calculation shows ∇2 g,adeg adeg = ∇2 adeg,g adeg = = (m2 0 m5 d cos2 x1) ∂ ∂x1 − (m2 0 m5 d cos2 x1) ∂ ∂x2 , thus ∇2 g,adeg adeg − ∇2 adeg,g adeg = 0 ∈ E0 satisfies (MF5)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The system is MF- linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A linearizing function is h(x) = md+J2 J2 x1 + x2 (all others giving MF-linearization are of the form σ h(x), where σ ∈ R\\ {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Due to the proof of Theorem 2, the linearizing diffeomorphism is (˜x, ˜y) = Φ(x, y) = (φ(x), Dφ(x)y) with φ(x) = (h, Leh)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The system in new coordinates reads ˙˜x1 = md + J2 J2 y1 + y2 = ˜y1 ˙˜y1 = md + J2 J2 �m0 md sin x1 − 1 md u � − m0 md sin x1 + md + J2 m2J2 u = m0 J2 sin x1 = Leh = ˜x2 (20) ˙˜x1 = m0 J2 cos x1y1 = ˜y2 ˙˜y2 = −m0 J2 sin x1y1y1 + m2 0 2mdJ2 sin(2x1) − m0 mdJ2 cos x1u = ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Example 3: We will study MF-linearizability of the TORA3 system (see Figure 1), which is based on the TORA system (Translational Oscillator with 16 Figure 1: The TORA3 system Rotational Actuator) studied in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' [19] (however we add gravita- tional effects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It consists of a two dimensional spring-mass system, with masses m1, m2 and spring constants k1, k2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A pendulum of length l3, mass m3, and moment of inertia J3 is added to the second body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The displacements of the bodies are denoted by x1 and x2, respectively, and the angle of the pen- dulum by x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The gravitational constant is a and u is a torque applied to the pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The kinetic energy is T =1 2m1( ˙x1)2 + 1 2(m2 + m3)( ˙x2)2 + 1 2(J3 + m3l2 3)( ˙x3)2 + m3l3 cos x3 ˙x2 ˙x3, and the mass matrix depends on the configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The potential energy is V = 1 2k1(x1)2 + 1 2k2(x2 − x1)2 − m3l3a cos x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The equations of the dynamics read m1¨x1 + k1x1 − k2 � x2 − x1� = 0 (m2 + m3)¨x2 + m3l3 cos x3¨x3 − m3l3 sin x3( ˙x3)2 +k2 � x2 − x1� = 0 m3l3 cos x3¨x2 + (m3l2 3 + J3)¨x3 + m3l3a sin x3 = u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' which can be rewritten on TQ as ˙x1 = y1 ˙y1 = η1 ˙x2 = y2 ˙y2 = −¯Γ2 33y3y3 + η2 + τ 2u ˙x3 = y3 ˙y3 = −¯Γ3 33y3y3 + η3 + τ 3u (21) where ¯Γ2 33 = −ν0 sin x3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ¯Γ3 33 = ν2 sin x3 cos x3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' η1 = − k1 m1 x1 + k2 m3 � x2 − x1� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' η2 = 1 2 ν2a sin 2x3−ν3(x2−x1) ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' η3 = ν4(x2−x1) cos x3−ν5 sin x3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' τ 2 = −m3l3 cos x3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' τ 3 = m2+m3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' with constant parameters: ν0 = m3l3(m3l2 3 + J3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ν1 = m2m3l2 3 + J3(m2 + m3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ν2 = m2 3l2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ν3 = k2 � m3l2 3 + J3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ν4 = m3l3k2 ν5 = m3l3a(m2 + m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 17 -- m1 m2 ki k2 W -- W aTo simplify calculations we apply to the system a preliminary mechanical feedback1 u = 1 τ 3 �¯Γ3 33y3y3 − η3 + ¯u � which yields ˙x1 = y1 ˙x2 = y2 ˙x3 = y3 ˙y1 = −µ1x1 + µ2x2 ˙y2 = µ3 sin x3y3y3 + µ4(x1 − x2) − µ3 cos x3u ˙y3 = ¯u, (22) with µ1 = k1+k2 m1 , µ2 = k2 m1 , µ3 = m3l3 m2+m3 , µ4 = k2 m2+m3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Since conditions (MF1)-(MF4) of Theorem 1 are MF-invariant, we will check them for system (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To summarize: Γ2 33 = −µ3 sin x3, and Γi jk = 0 otherwise, e = � −µ1x1 + µ2x2� ∂ ∂x1 + µ4 � x1 − x2� ∂ ∂x2 g = −µ3 cos x3 ∂ ∂x2 + ∂ ∂x3 = g2 ∂ ∂x2 + ∂ ∂x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We have (notice that calculations are performed on Q only) adeg = � µ2µ3 cos x3� ∂ ∂x1 − � µ3µ4 cos x3� ∂ ∂x2 , ad2 eg = µ3 cos x3 � (µ1µ2 + µ2µ4) ∂ ∂x1 − � µ2µ4 + µ2 4 � ∂ ∂x2 � , therefore rank E2 = 3 for x3 ̸= ± π 2 , and (MF1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now [g, adeg] = − � µ2µ3 sin x3� ∂ ∂x1 + � ��3µ4 sin x3� ∂ ∂x2 ∈ E1 and (MF2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then, for any vector field v = vi(x) ∂ ∂xi , ∇vg = �∂g2 ∂x3 + Γ2 33 � v3 ∂ ∂x2 = 0, thus (MC3) is satisfied if we replace v by, in particular, g, adeg, ad2 eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Finally, for (MF4), we calculate ∇2 g,ge = � µ2µ3 sin x3� ∂ ∂x1 − � µ3µ4 sin x3� ∂ ∂x2 ∈ E1, ∇2 adkeg,adj ege = 0 otherwise, thus, the system is MF-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, choose h = µ4 µ2 x1 + x2 + µ3 sin x3 (whose differential dh annihilates g and adeg), thus we take a linearizing diffeo- morphism (˜x, ˜y) = � φ(x), ∂φ ∂x(x)y � , with φ(x) = � h, Leh, L2 eh �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The linearized 1This preliminary feedback is not necessary and it is possible to check the conditions and to linearize the system without it, since our method and conditions are feedback invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 18 system is in the form of (LMS) and reads ˙˜x1 = µ4 µ2 y1 + y2 + µ3 cos x3y3 = ˜y1 ˙˜y1 = µ4 µ2 ˙y1+ ˙y2+µ3(cos x3 ˙y3−sin x3y3y3)= µ4(µ2 − µ1) µ2 x1 = ˜x2 ˙˜x2 = µ4(µ2 − µ1) µ2 y1 = ˜y2 ˙˜y2 = µ4(µ2 − µ1) µ2 ˙y1 = µ4(µ2 − µ1) µ2 � µ2x2 − µ1x1� = ˜x3 ˙˜x3 = µ1µ4(µ1 − µ2) µ2 y1 + µ4(µ2 − µ1)y2 = ˜y3 ˙˜y3 = (µ2 − µ1)µ3µ4 sin x3y3y3 − (µ1 − µ2)(µ2 1 + µ2µ4)µ4 µ2 x1 + (µ1 − µ2)(µ1 + µ4)µ4x2 + (µ1 − µ2)µ3µ4 cos x3u = ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 5 Conclusions In this paper, we consider MF-linearization of mechanical control systems (MS) with scalar control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We formulate the problem as a particular case of feedback linearization preserving the mechanical structure of (MS) so that the trans- formed system is both linear and mechanical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' As we showed in [4] and confirmed in this paper, even in the simplest case, the class of MF-linearizable systems is substantially smaller than that of general F-linearizable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore, a nat- ural question arises, namely to compare the conditions presented in this paper with those for F-linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The answer lies in the interplay between the distributions Ei = span � adj eg, 0 ≤ j ≤ i � and the ”usual” for F-linearization Di = span � adj F G, 0 ≤ j ≤ i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will address this problem in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 6 Appendix The following lemma can be proved by a direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The second covariant derivative ∇2 X,Y Z satisfies the following prop- erties: (i) linearity over C∞(Q) in X and Y : ∇2 (α1X1+α2X2),Y Z = α1∇2 X1,Y Z + α2∇2 X2,Y Z ∇2 X,(α1Y1+α2Y2)Z = α1∇2 X,Y1Z + α2∇2 X,Y1Z (ii) linearity over R in Z: ∇2 X,Y (a1Z1 + a2Z2) = a1∇2 X,Y Z1 + a2∇2 X,Y Z2 19 (iii) the product rule: ∇2 X,Y (βZ) =β∇2 X,Y Z + LXβ∇Y Z + LY β∇XZ + � ∇2 X,Y β � Z, where ∇2 X,Y β = LXLY β−L∇XY β ∈ C∞(Q), Xi, Yi, Zi ∈ X(Q), αi, β ∈ C∞(Q), and ai ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The following lemma is crucial for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For the system ˙x1 = y1 ˙xi = yi ˙y1 = u ˙yi = −Γi jkyjyk + xi−1, 2 ≤ i ≤ n, (23) we have for any 1 ≤ k, j ≤ n, ∇2 adk−1 e g,adj−1 e ge = (−1)j+k �∂Γi js ∂xk es + Γi jk+1 + Γi kj+1 − Γi−1 kj + (Γd jsΓi kd − Γd kjΓi ds)es � ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (24) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For system (23) we calculate ∇2 adk−1 e g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adj−1 e ge = (−1)j+k∇2 ∂ ∂xk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂ ∂xj e = ∇ ∂ ∂xk ∇ ∂ ∂xj e − ∇∇ ∂ ∂xk ∂ ∂xj e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where ∇ ∂ ∂xj e = � ∂ed ∂xj + Γd jses� ∂ ∂xd ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and ∇ ∂ ∂xk � ∇ ∂ ∂xj e � = ∇ ∂ ∂xk � ∂ed ∂xj � ∂ ∂xd + ∇ ∂ ∂xk � Γd jses� ∂ ∂xd = ∂ed ∂xj ∇ ∂ ∂xk ∂ ∂xd + L ∂ ∂xk � ∂ed ∂xj � ∂ ∂xd + � Γd jses� ∇ ∂ ∂xk ∂ ∂xd + � L ∂ ∂xk � Γd js � es + L ∂ ∂xk (es) Γd js � ∂ ∂xd = ∂ed ∂xj Γi kd ∂ ∂xi + Γd jsesΓi kd ∂ ∂xi + � ∂Γi js ∂xk es + ∂es ∂xk Γi js � ∂ ∂xi = � ∂Γi js ∂xk es + Γi jk+1 + Γi kj+1 + Γd jsΓi kdes � ∂ ∂xi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' since ∂ed ∂xj = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' if d = j + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and zero otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and thus ∂ed ∂xj Γi kd = Γi kj+1 (analogously for the other derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, using ∇ ∂ ∂xk ∂ ∂xj = Γd kj ∂ ∂xd , we calculate ∇∇ ∂ ∂xk ∂ ∂xj e = ∇Γd kj ∂ ∂xd e = Γd kj � ∂ei ∂xd + Γi dses � ∂ ∂xi = � Γi−1 kj + Γd kjΓi dses� ∂ ∂xi , so we have 20 ∇2 ∂ ∂xk , ∂ ∂xj e = ∇ ∂ ∂xk � ∇ ∂ ∂xj e � − ∇∇ ∂ ∂xk ∂ ∂xj e = �∂Γi js ∂xk es + Γi jk+1 + Γi kj+1 − Γi−1 kj + (Γd jsΓi kd − Γd kjΓi ds)es � ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' which yields (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Brockett, ”Feedback invariants for nonlinear systems”, in Proc.' metadata={'source': 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