Journal of Robotics

Journal of Robotics / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 892801 | 16 pages | https://doi.org/10.1155/2009/892801

A Riemannian-Geometry Approach for Modeling and Control of Dynamics of Object Manipulation under Constraints

Academic Editor: Heinz WΓΆrn
Received27 Oct 2008
Accepted27 Jan 2009
Published24 Mar 2009

Abstract

A Riemannian-geometry approach for modeling and control of dynamics of object manipulation under holonomic or non-holonomic constraints is presented. First, position/force hybrid control of an endeffector of a multijoint redundant (or nonredundant) robot under a holonomic constraint is reinterpreted in terms of β€œsubmersion” in Riemannian geometry. A force control signal constructed in the image space of the constraint gradient is regarded as a lifting (or pressing) in the direction orthogonal to the kernel space. By means of the Riemannian distance on the constraint submanifold, stability of position control under holonomic constraints is discussed. Second, modeling and control of two-dimensional object grasping by a pair of multijoint robot fingers are challenged, when the object is of arbitrary shape. It is shown that rolling contact constraints induce the Euler equation of motion, in which constraint forces appear as wrench vectors affecting the object. The Riemannian metric is introduced on a constraint submanifold characterized with arclength parameters. An explicit form of the quotient dynamics is expressed in the kernel space with accompaniment of a pair of first-order differential equations concerning the arclength parameters. An extension of Dirichlet-Lagrange's stability theorem to redundant systems under constraints is suggested by introducing a Morse-Lyapunov function.

1. Introduction

Among roboticsists, it is implicitly known that robot motions can be interpreted in terms of orbits on a high-dimensional torus or trajectories in an 𝑛-dimensional configuration space. Planning of robot motions has been investigated traditionally on the basis of kinematics on a configuration space as an 𝑛-dim numerical space 𝐑𝑛 [1].

This paper first emphasizes a mathematical observation that, given a robot as a multibody mechanism with n degrees of freedom whose endpoint is free, the set of all its postures can be regarded as a Riemannian manifold (𝑀, 𝑔) associated with the Riemannian metric 𝑔 that constitutes the robot inertia matrix. A geodesic connecting any two postures can correspond to an orbit expressed on a local coordinate chart and generated by a solution to the Euler-Lagrange equation of robot motion that originates only from the force of inertia [2, 3]. It should be emphasized that once the Riemannian manifold is given corresponding to the 𝑛 degrees of freedom robot, the collection of all the geodesic paths describes the β€œlaw of inertia” for the manifold. It is also important to note that geodesic paths are invariant under any choice of local coordinates. This Riemannian geometry viewpoint is extended in this paper to an important class of multibody dynamics physically interacting with an object or with environment through holonomic or/and nonholonomic (but Pfaffian) constraints. Holonomic constraints are defined as a set of infinitely differentiable functions from a product manifold of multibody Riemannian manifolds onto an open set of a 2- or 3-dimensional Euclidean space called the task space. Such a mapping can be treated as a submersion from the product Riemannian manifold to π‘š(=2-or3-) dimensional Euclidean space. Hence, holonomic constraints induce a Riemannian submanifold with a naturally induced metric. An Euler-Lagrange equation is formulated in an implicit function form under such constraints. It is also shown that if the gravity term can be explicitly compensated and there arises no viscous friction then the geodesic motion is invariant, that is, it is governed by the β€œlaw of inertia,” under any adequate lifting (or pressing) through the joint torque injection in the direction along the constraint gradient. An explicit form of the Euler equation whose solution corresponds to a geodesic on the submanifold is given also as a quotient dynamics corresponding to the kernel space as an orthogonal compliment to the image space spanned from all the constraint gradients. Based upon these observations, the well-known methodology of hybrid (position/force) control for a robot whose end effector is constrained on a surface is re-examined and shown to be effective even if the robot is of redundancy in its degrees of freedom.

In a latter part of the paper, modeling of dynamics of grasping and manipulation of a two-dimensional rigid object with arbitrary shape by using a pair of multijoint robot fingers with spherical finger ends is challenged. It is shown that rolling contact constraints between finger ends and the object surfaces induce not only two holonomic constraints but also two nonholonomic constraints that restrict tangent vectors on the original Riemannian manifold that is a product of three manifolds expressing a set of whole postures of the two fingers and the object. An Euler-Lagrange equation for expressing the dynamics of such physical interaction is derived through applying the variational principle together with deriving a set of the first-order differential equations expressing the contact positions of the object with both the finger ends. The Riemannian distance is introduced on the kernel space as an orthogonal compliment to the image space of all the gradients vectors of both contact and rolling constraints. In other words, rolling constraints are expressed in terms of the first fundamental forms of given contours of the object and restrict only the tangent vector fields at both the contact points. An explicit Euler-Lagrange equation corresponding to a path on the constraint submanifold is derived together with a set of the first-order differential equations expressed in terms of the second fundamental forms of the object contours. Thus, it is shown that rolling constraints can be characterized by means of arc length parameters of the object contours that express locations of the contact points and in the sequel are integrable in the sense of Frobenius. A coordinated control signal called β€œblind grasping” without referring to the object kinematics or external sensing is proposed and shown to be effective in realizing stable grasping in the sense of stability on a submanifold. A sketch of the convergence proof is given on the basis of an extension of the Dirichlet-Lagrange theorem to a system of degrees of freedom redundancy by finding a Morse-Lyapunov function and using its physical properties and mathematical meanings.

2. Riemannian Manifold: A Set of All Postures

Lagrange's equation of motion of a multijoint system with 2 degrees of freedom (DOF) shown in Figure 1 is described by the formula𝐻(π‘ž)Μˆπ‘ž+12̇𝐻(π‘ž)+𝑆(π‘ž,Μ‡π‘ž)ξ‚‡Μ‡π‘ž+𝑔(π‘ž)=𝑒,(1)where π‘ž=(π‘ž1,π‘ž2)𝑇 denotes the vector of joint angles, 𝐻(π‘ž) denotesthe inertia matrix, 𝑆(π‘ž,Μ‡π‘ž)Μ‡π‘ž the gyroscopic force term including centrifugal and Coriolis forces, 𝑔(π‘ž) the gradient vector of a potential function 𝑃(π‘ž) due to the gravity with respect to π‘ž, and 𝑒 the joint torque generated by joint actuators [4]. It is well known that the inertia matrix 𝐻(π‘ž) is symmetric and positive definite, and there exist a positive constant β„Žπ‘š together with a positive definite constant diagonal matrix 𝐻0 such thatβ„Žπ‘šπ»0≀𝐻(π‘ž)≀𝐻0(2)for any π‘ž. It should be also noted that 𝑆(π‘ž,Μ‡π‘ž) is skew symmetric and linear and homogeneous in Μ‡π‘ž. More in detail, the 𝑖𝑗th entry of 𝑆(π‘ž,Μ‡π‘ž) denoted by 𝑠𝑖𝑗 can be described in the form [3]𝑠𝑖𝑗=12ξ‚»πœ•πœ•π‘žπ‘—ξ‚΅π‘›ξ“π‘˜=1Μ‡π‘žπ‘˜β„Žπ‘–π‘˜ξ‚Άβˆ’πœ•πœ•π‘žπ‘–ξ‚΅π‘›ξ“π‘˜=1Μ‡π‘žπ‘˜β„Žπ‘—π‘˜ξ‚Άξ‚Ό,(3)where 𝐻(π‘ž)=(β„Žπ‘–π‘—(π‘ž)), from which it follows apparently that 𝑠𝑖𝑗=βˆ’π‘ π‘—π‘–. Since we assume that the objective system to be controlled is a series of rigid links serially connected through each rotational joint with single DOF, every entry of 𝐻(π‘ž) is a constant or a sinusoidal function of components of joint angle vector π‘ž. That is, each element of 𝐻(π‘ž) and 𝑔(π‘ž) is differentiable of class 𝐢∞ (infinitely differentiable in π‘ž).

When two joint angles πœƒ1 and πœƒ2 are given in πœƒπ‘–βˆˆ(βˆ’πœ‹,πœ‹], 𝑖=1,2, for the 2 DOF robot arm shown in Figure 1, the posture 𝑝(πœƒ1,πœƒ2) is determined naturally. Denote the set of all such possible postures by 𝑀 and introduce a family of subsets of 𝑀 such that, for any π‘βˆˆπ‘€ with joint angles 𝑝=(πœƒ1,πœƒ2) and any number 𝛼>0, a set of all 𝑝′=(πœƒξ…ž1,πœƒξ…ž2) is defined asπ‘ˆπ‘,𝛼=ξ€½π‘ξ…žβˆΆβ€–β€–π‘ξ…žβˆ’π‘β€–β€–π»<𝛼,(4)whereβ€–β€–π‘ξ…žβˆ’π‘β€–β€–π»=ξƒŽξ“π‘–,π‘—β„Žπ‘–π‘—(𝑝)ξ€·πœƒξ…žπ‘–βˆ’πœƒπ‘–ξ€Έξ€·πœƒξ…žπ‘—βˆ’πœƒπ‘—ξ€Έ(5)can be regarded as an open subset of 𝑀. Then, the set 𝑀 with such a family of open subsets can be regarded as a topological manifold. It is possible to show that the manifold 𝑀 becomes Hausdorff and compact. Further, every point 𝑝 of 𝑀 has a neighborhood π‘ˆ that is homeomorphic to an open subset Ξ© of 2-dimensional numerical space 𝐑2. Such a homeomorphism πœ™βˆΆπ‘ˆβ†’Ξ© is called a coordinate chart. In fact, a neighborhood π‘ˆπ‘,𝛼 of posture 𝑝 with joint angles (πœƒ1,πœƒ2) in Figure 1 can be mapped to an open set Ξ© in 𝐑2 with 2 dimensional numerical coordinates (π‘ž1,π‘ž2) with the origin 𝑂 (Figure 2). In this case, it is possible to see that the original set 𝑀 of robot postures can be visualized as a torus shown in 𝐑3 (see Figure 3) in which angles π‘ž1 and π‘ž2 are defined. It is quite fortunate to see that, in the case of typical robots like the one shown in Figure 1, the local coordinates (π‘ž1,…,π‘žπ‘›) can be identically chosen as a set of 𝑛 independent joint angles (πœƒ1,…,πœƒπ‘›) by setting π‘žπ‘–=πœƒπ‘–(𝑖=1,…,𝑛). It is also interesting to see that the torus in Figure 3 is made to be topologically coincident with the set of all arm endpoints 𝑃=(π‘₯,𝑦,𝑧). As discussed in detail in mathematical text books [5, 6], the topological manifold (𝑀,𝑝) of such a torus can be regarded as a differentiable manifold of class 𝐢∞.

Now, it is necessary to define a tangent vector to an abstract differentiable manifold 𝑀 at π‘βˆˆπ‘€. Let 𝐼 be an interval (βˆ’πœ–,πœ–) and define a curve 𝑐(𝑑) by a mapping π‘βˆΆπΌβ†’π‘€ such that 𝑐(0)=𝑝. A tangent vector to 𝑀 at 𝑝 is an equivalence class of curves π‘βˆΆπΌβ†’π‘€ for the equivalence relation ∼ defined by

π‘βˆΌπ‘if and only if, in a coordinate chart (π‘ˆ,πœ™) around 𝑝, we have (πœ™βˆ˜π‘)ξ…ž(0)=(πœ™βˆ˜π‘)ξ…ž(0), where symbol ()β€² means differentiation of () with respect to π‘‘βˆˆπΌ. This definition of tangent vectors to 𝑀 at 𝑝 does not depend on choice of the coordinate chart at 𝑝, as discussed in text books [5, 6]. Let us denote the set of all tangent vectors to 𝑀 at 𝑝 by 𝑇𝑝𝑀 and call it the tangent space at π‘βˆˆπ‘€. It has an 𝑛-dimensional linear space structure like 𝐑𝑛. We also denote the disjoint union of the tangent spaces to 𝑀 at all the points of 𝑀 by 𝑇𝑀 and call it the tangent bundle of 𝑀.

Now, we are in a position to define a Riemannian metric on a differentiable manifold (𝑀,𝑝) as a mapping π‘”π‘βˆΆπ‘‡π‘π‘€Γ—π‘‡π‘π‘€β†’π‘ such that 𝑝→𝑔𝑝 is of class 𝐢∞ and 𝑔𝑝(𝑒,𝑣) for π‘’βˆˆπ‘‡π‘π‘€ and π‘£βˆˆπ‘‡π‘π‘€ is a symmetric positive definite quadratic form𝑔𝑝(𝑒,𝑣)=𝑛𝑖,𝑗=1𝑔𝑖𝑗(𝑝)𝑒𝑖𝑣𝑗.(6)

Suppose that 𝑀 is a connected Riemannian manifold. If π‘βˆΆπΌ[π‘Ž,𝑏]→𝑀 is a curve segment of class 𝐢∞, we define the length of 𝑐 to beξ€œπΏ(𝑐)=π‘π‘Žβ€–β€–β€–β€–=ξ€œΜ‡π‘(𝑑)dπ‘‘π‘π‘Žξ”π‘”π‘(𝑑)̇𝑐(𝑑),̇𝑐(𝑑)d𝑑,(7)where we assume ̇𝑐(𝑑)β‰ 0 for any π‘‘βˆˆπΌ and call such a curve segment to be regular. A mapping of class πΆβˆžπ‘βˆΆ[π‘Ž,𝑏]→𝑀 is called a piecewise regular curve segment (for brevity, we call it an admissible curve) if there exists a finite subdivision π‘Ž=π‘Ž0<π‘Ž1<β‹―<π‘Žπ‘˜=𝑏 such that 𝑐(𝑑) for π‘‘βˆˆ[π‘Žπ‘–βˆ’1,π‘Žπ‘–] is a regular curve for 𝑖=1,…,π‘˜. Then, it is possible to define for any pair of points 𝑝,π‘β€²βˆˆπ‘€ the Riemannian distance 𝑑(𝑝,π‘ξ…ž) to be the infimum of all admissible curves from 𝑝 to π‘ξ…ž. It is well known [4, 5] that, with the distance function 𝑑 defined above, any connected Riemannian manifold becomes a metric space whose induced topology is coincident with the given manifold topology. An admissible curve 𝑐 in a Riemannian manifold is said to be minimizing if 𝐿(𝑐)≀𝐿(̃𝑐) for any other admissible curve ̃𝑐 with the same endpoints. It follows immediately from the definition of distance that 𝑐 is minimizing if and only if 𝐿(𝑐) is equal to the distance between its endpoints. Further, it is known that if the Riemannian manifold {𝑀,𝑔} is complete, then for any pair of points 𝑝 and π‘ξ…ž there exists at least a minimizing curve 𝑐(𝑑), π‘‘βˆˆ[π‘Ž,𝑏], with 𝑐(π‘Ž)=𝑝 and 𝑐(𝑏)=π‘ξ…ž. If such a minimizing curve 𝑐(𝑑) is described with the aid of coordinate chart (π‘ˆ,πœ™) as πœ™(𝑐(𝑑))=(π‘ž1(𝑑),…,π‘žπ‘›(𝑑)), then π‘ž(𝑑)=πœ™(𝑐(𝑑)) satisfies the 2nd-order differential equationd2d𝑑2π‘žπ‘˜(𝑑)+𝑛𝑖,𝑗=1Ξ“π‘˜π‘–π‘—(𝑐(𝑑))dπ‘žπ‘–(𝑑)d𝑑dπ‘žπ‘—(𝑑)d𝑑=0,(π‘˜=1,…,𝑛),(8)where Ξ“π‘˜π‘–π‘— denotes Christoffel's symbol defined byΞ“π‘˜π‘–π‘—=12π‘›ξ“β„Ž=1π‘”π‘˜β„Žξ‚΅πœ•π‘”π‘–β„Žπœ•π‘žπ‘—+πœ•π‘”π‘—β„Žπœ•π‘žπ‘–βˆ’πœ•π‘”π‘–π‘—πœ•π‘žβ„Žξ‚Ά,(9)and (π‘”π‘˜β„Ž) denotes the inverse of matrix (π‘”π‘˜β„Ž). A curve π‘ž(𝑑)βˆΆπΌβ†’π‘ˆ satisfying (8) together with πœ™βˆ’1(π‘ž(𝑑)) is called a geodesic, and (8) itself is called the Euler-Lagrange equation or the geodesic equation.

Given a 𝐢∞-class curve 𝑐(𝑑)=𝐼[π‘Ž,𝑏]→𝑀, the quantity1𝐸(𝑐)=2ξ€œπ‘π‘Žβ€–β€–β€–β€–Μ‡π‘(𝑑)2=1d𝑑2ξ€œπ‘π‘Žπ‘”π‘(𝑑)̇𝑐(𝑑),̇𝑐(𝑑)d𝑑(10)is called the energy of the curve. Then, by applying the Cauchy-Schwartz inequality for (7), we have𝐿(𝑐)2≀2(π‘βˆ’π‘Ž)𝐸(𝑐).(11)Further, the equality of (11) follows if and only if ‖̇𝑐(𝑑)β€– is constant. It is also possible to see that if 𝑐(𝑑) is a geodesic with 𝑐(π‘Ž)=𝑝 and 𝑐(𝑏)=π‘ξ…ž, then for any other curve 𝑐(𝑑) with the same endpoints, it holds𝐸(𝑐)=𝐿(𝑐)2≀𝐿2(π‘βˆ’π‘Ž)𝑐2ξ€·2(π‘βˆ’π‘Ž)≀𝐸𝑐.(12)The equalities hold if and only if 𝑐(𝑑) is also a geodesic. Conversely, if 𝑐(𝑑) with 𝑐(π‘Ž)=𝑝 and 𝑐(𝑏)=π‘ξ…ž is a 𝐢∞ curve that minimizes the energy and makes 𝑔𝑐(𝑑)(̇𝑐(𝑑),̇𝑐(𝑑)) constant, then 𝑐(𝑑) becomes a geodesic connecting 𝑐(π‘Ž)=𝑝 and 𝑐(𝑏)=π‘ξ…ž. In mechanics, 𝐸(𝑐) is usually called β€œaction of 𝑐,” and 𝑐(𝑑) is considered as the orbit of motion of a multibody system.

3. Riemannian Geometry of Robot Dynamics

Dynamics of a robotic mechanism with 𝑛 rigid bodies connected in series through rotational joints are described by Lagrange's equation of motion, as shown in (1). It is implicitly assumed that the axis of rotation of the first body is fixed in an inertial frame and denoted by 𝑧-axis that is perpendicular to the π‘₯𝑦-plane as shown in Figure 1. If there is no gravity force affecting motion of the robot, then the equation of motion of the robot can be described by the form1𝐻(π‘ž)Μˆπ‘ž+2̇𝐻(π‘ž)+π‘†π‘ž,Μ‡π‘žΜ‡π‘ž=𝑒,(13)where 𝑒 stands for a vector of control torques emanating from joint actuators. This formula is valid for motions of a revolute joint robot, shown in Figure 1, if it is installed in weightless environment like an artificial satellite, or the gravity term 𝑔(π‘ž) (included in (1)) can be compensated by joint actuators through control input 𝑒. In general, we can represent a posture 𝑝 of the robot as a physical entity by a family of joint angles πœƒπ‘–(𝑖=1,…,𝑛), which can be expressed by a point Θ=(πœƒ1,…,πœƒπ‘›) in the 𝑛-dimensional numerical space 𝐑𝑛. In fact, we can naturally imagine an isomorphism πœ™βˆΆπ‘ˆβ†’Ξ©, where π‘ˆβŠ‚π‘€, and Ξ© is an open subset of 𝐑𝑛. In other words, a local coordinate chart πœ™(π‘ˆ)(=Ξ©) can be treated to be identical to π‘ˆ itself, an open subset of 𝑀, by regarding π‘ž=(π‘ž1,…,π‘žπ‘›)𝑇 (β€œπ‘‡β€ denotes transpose and hence π‘ž a column vector) identical to Θ by setting π‘žπ‘–=πœƒπ‘–(𝑖=1,…,𝑛). In this way, the abstract manifold 𝑀 as the set of all robot postures can be regarded as an 𝑛-dimensional torus 𝑇𝑛 as an 𝑛-tuple direct product of 𝑆1βˆΆπ‘‡π‘›=𝑆1×⋯×𝑆1. Hence, a robot posture π‘βˆˆπ‘€ can be represented by a point Θ on 𝑇𝑛 and also expressed by a joint vector π‘ž in 𝐑𝑛.

From the definition of inertia matrices, 𝐻(π‘ž) is symmetric and positive definite, and the kinetic energy is expressed as a quadratic form𝐾=1π‘ž,Μ‡π‘ž2Μ‡π‘žπ‘‡π»(π‘ž)Μ‡π‘ž.(14)Hence, the equation of motion of the robot is expressed by Lagrange's equationdξ‚†πœ•dπ‘‘πΏξ€·ξ€Έξ‚‡βˆ’πœ•πœ•Μ‡π‘žπ‘ž,Μ‡π‘žπΏξ€·ξ€Έπœ•π‘žπ‘ž,Μ‡π‘ž=𝑒,(15)where 𝐿(π‘ž,Μ‡π‘ž)=𝐾(π‘ž,Μ‡π‘ž), and 𝑒 stands for a generalized external force vector. It is interesting to note that in differential geometry, (15) can be described asξ“π‘–β„Žπ‘˜π‘–Μˆπ‘žπ‘–+𝑖,π‘—Ξ“π‘–π‘˜π‘—(π‘ž)Μ‡π‘žπ‘–Μ‡π‘žπ‘—=π‘’π‘˜,(16)where Ξ“π‘–π‘˜π‘— denotes Christoffel's symbol of the first kind defined byΞ“π‘–π‘˜π‘—=12ξ‚΅πœ•β„Žπ‘—π‘˜πœ•π‘žπ‘–+πœ•β„Žπ‘–π‘˜πœ•π‘žπ‘—βˆ’πœ•β„Žπ‘–π‘—πœ•π‘žπ‘˜ξ‚Ά.(17)For later use, we introduce another Christoffel's symbol called the second kind as shown in the formulaΞ“π‘˜π‘–π‘—=12𝑛𝑙=1β„Žπ‘™π‘˜ξ‚΅πœ•β„Žπ‘—π‘™πœ•π‘žπ‘–+πœ•β„Žπ‘–π‘™πœ•π‘žπ‘—βˆ’πœ•β„Žπ‘–π‘—πœ•π‘žπ‘™ξ‚Ά=12𝑛𝑙=1β„Žπ‘™π‘˜Ξ“π‘–π‘™π‘—,(18)where (β„Žπ‘™π‘˜) denotes the inverse of (β„Žπ‘™π‘˜), the inertia matrix 𝐻(π‘ž)=(β„Žπ‘™π‘˜). Since (β„Žπ‘˜π‘™) and (β„Žπ‘˜π‘™) are symmetric, it follows that Ξ“π‘–π‘˜π‘—=Ξ“π‘—π‘˜π‘– and Ξ“π‘˜π‘–π‘—=Ξ“π‘˜π‘—π‘–. Now, we show that (13) is equivalent to (16) by bearing in mind that Μ‡βˆ‘π»(π‘ž)=𝑖{πœ•π»(π‘ž)/πœ•π‘žπ‘–}Μ‡π‘žπ‘–, and the skew symmetric matrix 𝑆(π‘ž,Μ‡π‘ž) is expressed as in (3). In fact, the second term in the bracket () of (17) corresponds to the first term in {} of (3) and the third term of (17) does to the second term in{} of (3). Hence, it follows from (3) that𝑛𝑗=1π‘ π‘˜π‘—Μ‡π‘žπ‘—=𝑛𝑗=112πœ•ξ‚Έξ‚»πœ•π‘žπ‘—ξ‚΅π‘›ξ“π‘–=1Μ‡π‘žπ‘–β„Žπ‘˜π‘–ξ‚Άξ‚ΌΜ‡π‘žπ‘–βˆ’ξ‚»πœ•πœ•π‘žπ‘˜ξ‚΅π‘›ξ“π‘–=1Μ‡π‘žπ‘–β„Žπ‘–π‘—ξ‚Άξ‚ΌΜ‡π‘žπ‘—ξ‚Ή=𝑛𝑛𝑗=1𝑖=112ξ‚»ξ‚΅πœ•β„Žπ‘–π‘˜πœ•π‘žπ‘—βˆ’πœ•β„Žπ‘–π‘—πœ•π‘žπ‘˜ξ‚Άξ‚ΌΜ‡π‘žπ‘–Μ‡π‘žπ‘—.(19)Substituting this into (16) by comparing the last two terms of (17) with the last bracket {} of (19) results in the equivalence of (13) to (16). It is easy to see that multiplication of (16) by π»βˆ’1(π‘ž) yieldsΜˆπ‘žπ‘˜+𝑖,π‘—Ξ“π‘˜π‘–π‘—Μ‡π‘žπ‘–Μ‡π‘žπ‘—=ξ“π‘—β„Žπ‘˜π‘—π‘’π‘—,π‘˜=1,…,𝑛.(20)If 𝑒=0, this expression is nothing, but the Euler-Lagrange equation shown in (8). By this reason, from now on, we use symbol 𝑔𝑖𝑗(π‘ž) instead of β„Žπ‘–π‘—(π‘ž) for the inertia matrix 𝐻(π‘ž) even when robot dynamics are treated.

Now, on the abstract topological manifold 𝑀 as a set of all possible postures of a robot, suppose that a Riemannian metric is given by a scalar product on each tangent space 𝑇𝑝𝑀:βŸ¨π‘£,π‘€βŸ©=𝑔𝑖𝑗(𝑝)𝑣𝑖𝑀𝑗,(21)where 𝑣=𝑣𝑖(πœ•/πœ•π‘žπ‘–)βˆˆπ‘‡π‘π‘€ and 𝑀=𝑀𝑗(πœ•/πœ•π‘žπ‘—)βˆˆπ‘‡π‘π‘€, and the summation symbol βˆ‘ in 𝑖 and 𝑗 is omitted, and π‘ž=(π‘ž1,…,π‘žπ‘›) represents local coordinates. Then, the manifold (𝑀,𝑝) can be regarded as a Riemannian manifold and becomes complete as a metric space. Then, according to the Hopf-Rinow theorem [5], any two points 𝑝,π‘žβˆˆπ‘€ can be joined by a geodesic of length 𝑑(𝑝,π‘ž), that is, a curve satisfying (8) with shortest length, whereξ€œπ‘‘(𝑝,π‘ž)=infπ‘π‘Žβ€–β€–β€–β€–ξ€œΜ‡π‘(𝑑)d𝑑=infπ‘π‘Žξ”βŸ¨Μ‡π‘(𝑑),̇𝑐(𝑑)⟩d𝑑(22)with 𝑐(π‘Ž)=𝑝 and 𝑐(𝑏)=π‘ž.

As discussed in the previous section, geodesics are the critical points of the energy functional 𝐸(𝑐). Further, a geodesic curve 𝑐(𝑑) satisfies ‖̇𝑐(𝑑)β€–=const. In fact, by regarding 𝑐(𝑑)=π‘ž(𝑑) that is an orbit on Ξ©, we havedddπ‘‘βŸ¨Μ‡π‘ž,Μ‡π‘žβŸ©=𝑔dπ‘‘π‘–π‘—ξ€·ξ€Έπ‘ž(𝑑)Μ‡π‘žπ‘–(𝑑)Μ‡π‘žπ‘—ξ€Έ=d(𝑑)ξ€½dπ‘‘Μ‡π‘žπ‘‡ξ€Ύ(𝑑)𝐺(π‘ž(𝑑))Μ‡π‘ž(𝑑)=Μˆπ‘žπ‘‡(𝑑)𝐺(π‘ž)Μ‡π‘ž(𝑑)+Μ‡π‘žπ‘‡Μ‡(𝑑)𝐺(π‘ž)Μ‡π‘ž(𝑑)+Μ‡π‘žπ‘‡(𝑑)𝐺(π‘ž)Μˆπ‘ž(𝑑)=0,(23)where the equivalent expression1𝐺(π‘ž)Μˆπ‘ž+2̇𝐺(π‘ž)+π‘†π‘ž,Μ‡π‘žΜ‡π‘ž=0(24)to (13) with 𝑒=0 is used, 𝐺(π‘ž)=(𝑔𝑖𝑗(π‘ž)), and 𝑠𝑖𝑗 of 𝑆 is given in the form of (3) (where β„Žπ‘–π‘—=𝑔𝑖𝑗). It is also important to note that, on a local coordinate chart Ξ©βŠ‚π‘π‘› corresponding to a neighborhood π‘ˆ of π‘βˆˆπ‘€, an orbit π‘ž(𝑑) parameterized by time π‘‘βˆˆ[π‘Ž,𝑏] and expressed by a solution to (20) (where 𝑒𝑗 is of 𝐢∞ in 𝑑) should satisfy𝑛𝑗=1Μ‡π‘žπ‘—(𝑑)𝑒𝑗(𝑑)=dd𝑑𝐸(π‘ž(𝑑)),(25)orξ€œπ‘π‘Žπ‘›ξ“π‘—=1Μ‡π‘žπ‘—(𝑑)𝑒𝑗(𝑑)d𝑑=𝐸(π‘ž(𝑏))βˆ’πΈ(π‘ž(π‘Ž)),(26)as long as π‘ž(𝑑)∈Ω, where 𝐸(π‘ž(𝑑))=(1/2)βŸ¨Μ‡π‘ž(𝑑),Μ‡π‘ž(𝑑)⟩. When 𝑒(𝑑)=0, 𝐸(π‘ž(𝑑))=const. and then the curve connecting 𝑝=πœ™βˆ’1(π‘ž(π‘Ž)) and π‘ξ…ž=πœ™βˆ’1(π‘ž(𝑏)) must be a geodesic. In other words, an inertia-originated movement without being affected by the gravitational field or any external force field produces a geodesic orbit [2]. The most importantly, geodesics together with their length are invariant under any choice of local coordinates.

Before closing this expository section on robot motion from the Riemannian geometry viewpoint, we must emphasize that all the above invariant properties of geodesics of inertia-originated robot motions result from imaging a set of all robot postures as an abstract Riemannian manifold. Choice of a local coordinates is originally arbitrary. Even an 𝑛-dimensional torus 𝑇𝑛 is one of such choice of local coordinates corresponding to the choice of joint angles π‘ž=(π‘ž1,…,π‘žπ‘›)𝑇. At the same time, it is important to note that, in differential geometry, choice of coordinates in the tangent space 𝑇𝑝𝑀 is indeterminable or free to choose. However, once a local coordinates system for an 𝑛 degrees of freedom robot is chosen by joint angle vector π‘ž=(π‘ž1,…,π‘žπ‘›)𝑇, then the coordinates in the tangent space 𝑇𝑝𝑀 should be chosen as the vector of joint angular velocities πœ•/πœ•π‘ž=(πœ•/πœ•π‘ž1,…,πœ•/πœ•π‘žπ‘›) correspondingly to Μ‡π‘ž=(Μ‡π‘ž1,…,Μ‡π‘žπ‘›)𝑇, from which the Riemannian metric 𝑔𝑖𝑗(π‘ž) is defined through the inertia matrix.

4. Constraint Submanifold and Hybrid Position/Force Control

Consider an 𝑛-DOF robotic arm whose last link is a pencil and suppose that the endpoint of the pencil is in contact with a flat surface πœ‘(𝐱)=πœ‰, where 𝐱=(π‘₯,𝑦,𝑧)𝑇. It is well known that the Lagrange equation of motion of the system is written as𝐺(π‘ž)Μˆπ‘ž+12̇𝐺+π‘†ξ‚‡Μ‡π‘ž+𝑔(π‘ž)=βˆ’πœ†πœ•πœ‘(𝐱(π‘ž))πœ•π‘ž+𝑒,(27)where πœ•πœ‘/πœ•π‘ž can be decomposed into πœ•πœ‘(π‘ž)/πœ•π‘ž=𝐽𝑇(π‘ž)πœ•πœ‘/πœ•π± and 𝐽(π‘ž)=πœ•π±/πœ•π‘žπ‘‡. On the constraint manifold πΉπœ‰={π‘βˆΆπ‘βˆˆπ‘€andπœ‘(𝐱(𝑝))=πœ‰}, let us consider a smooth curve 𝑐(𝑑)∢𝐼[π‘Ž,𝑏]β†’πΉπœ‰ that connects the given two points 𝑐(π‘Ž)=𝑝 and 𝑐(𝑏)=π‘ξ…ž, where 𝑝 and π‘ξ…ž belong to πΉπœ‰. The length of such a curve constrained to πΉπœ‰ is defined as𝐿(𝑐)=ξ€œπ‘π‘Žξ”π‘”π‘–π‘—(𝑐(𝑑))̇𝑐𝑖(𝑑)̇𝑐𝑗(𝑑)d𝑑,(28)and consider the minimization𝑑𝑝,π‘ξ…žξ€Έ=infπ‘βˆˆπΉπœ‰πΏ(𝑐)(29)that should be called the distance between 𝑝 and π‘ξ…ž on the constraint manifold. Then, the minimizing curve called the geodesic denoted identically by π‘ž(𝑑)(=𝑐(𝑑)) must satisfy the Euler equationΜˆπ‘žπ‘˜(𝑑)+Ξ“π‘˜π‘–π‘—Μ‡π‘žπ‘–(𝑑)Μ‡π‘žπ‘—(𝑑)=βˆ’πœ†(𝑑)β‹…ξ€·gradπœ‘(𝐱(𝑑))ξ€Έπ‘˜,(30)together with the constraint condition πœ‘(𝐱(𝑑))=πœ‰, wheregradπœ‘(𝐱(𝑑))=πΊβˆ’1(π‘ž(𝑑))𝐽𝑇(π‘ž)πœ•πœ‘πœ•π±,(31)and 𝐽𝑇(π‘ž)=πœ•π±π‘‡/πœ•π‘ž. It should be noted that, from the inner product of (30) and 𝑀=π½π‘‡πœ•πœ‘/πœ•π±, it follows thatξ“π‘˜ξ€·π‘€π‘˜Μˆπ‘žπ‘˜+π‘€π‘˜Ξ“π‘˜π‘–π‘—Μ‡π‘žπ‘–Μ‡π‘žπ‘—ξ€Έ=βˆ’πœ†ξ“π‘˜π‘€π‘˜π‘”π‘˜π‘—π‘€π‘—.(32)Since the holonomic constraint πœ‘(𝐱(π‘ž))=πœ‰ implies βŸ¨π‘€,Μ‡π‘žβŸ©=0, it follows that4ξ“π‘˜=1ξ‚€π‘€π‘˜Μˆπ‘žπ‘˜+ξ‚€ddπ‘‘π‘€π‘˜ξ‚Μ‡π‘žπ‘˜ξ‚=0.(33)Substituting this into (32), we obtainπœ†(𝑑)=1π‘€π‘‡πΊβˆ’1π‘€ξ‚»ξ“π‘˜ξ‚΅Μ‡π‘€π‘˜Μ‡π‘žπ‘˜βˆ’ξ“π‘–,π‘—π‘€π‘˜Ξ“π‘˜π‘–π‘—Μ‡π‘žπ‘–Μ‡π‘žπ‘—ξ‚Άξ‚Ό.(34)From the Riemannian geometry, the constraint force πœ†(𝑑) with the grad{πœ‘(𝐱(π‘ž))} stands for a component of the image space of 𝑀(=𝐽𝑇(π‘ž)πœ•πœ‘/πœ•π±) that is orthogonal to the kernel π‘‡πΉπœ‰ of 𝑀. In other words, this component is cancelled out by the image space component of the left hand side of (32). From the physical point of view, πœ†(𝑑) should be regarded as a magnitude of the constraint force that presses the surface πœ‘(𝐱(π‘ž))=πœ‰ in its normal direction. In order to compromise the mathematical argument with such physical reality, let us suppose that the joint actuators can supply the control torques𝑒=πœ†π‘‘β‹…π½π‘‡(π‘ž)πœ•πœ‘πœ•π±+𝑔(π‘ž).(35)Then, by substituting this into (27), we obtain the Lagrange equation of motion under the constraint πœ‘(𝐱(π‘ž))=πœ‰:𝐺(π‘ž)Μˆπ‘ž+12̇𝐺+π‘†ξ‚‡Μ‡π‘ž=βˆ’ξ€·πœ†βˆ’πœ†π‘‘ξ€Έπ½π‘‡(π‘ž)πœ•πœ‘πœ•π±=βˆ’Ξ”πœ†π½π‘‡(π‘ž)πœ•πœ‘πœ•π±,(36)where Ξ”πœ†=πœ†βˆ’πœ†π‘‘. It should be noted that introduction of the first term of control signal of (35) does not affect the solution orbit on the constraint manifold, and further it keeps the constraint condition during motion by rendering πœ†(𝑑)(=πœ†π‘‘+Ξ”πœ†(𝑑)) positive. In a mathematical sense, exertion of the joint torque πœ†π‘‘π½π‘‡(πœ•πœ‘/𝐱) plays a role of β€œpressing” or β€œlifting” of the image space spanned from the gradient of the constraint equation. Further, note that (36) is of an implicit function form with the Lagrange multiplier Ξ”πœ†. To affirm the argument of treatment of the geodesics through this implicit form, we show an explicit form of the Lagrange equation expressed on the orthogonally projected space (kernel space) by introducing the orthogonal transformationΜ‡π‘ž=𝑃,π‘€β€–π‘€β€–βˆ’1ξ€Έξ‚΅Μ‡πœΌΜ‡π‘§ξ‚Ά=π‘„Μ‡π‘ž,(37)where 𝑃 is a 4Γ—3 matrix whose column vectors with the unit norm are orthogonal to 𝑀, and Μ‡πœΌ denotes a 3Γ—1 matrix (3-dim. vector) and ̇𝑧 a scalar. Since 𝑄 is an orthogonal matrix, π‘„βˆ’1=𝑄𝑇. Hence, if Μ‡π‘žβˆˆker(𝑀)(=π‘‡πΉπœ‰), then ̇𝑧=0. Restriction of (36) to the kernel space of 𝑀 can be attained by multiplying (36) by 𝑃𝑇 from the left such that𝑃𝑇𝐺(π‘ž)dd𝑑(π‘ƒΜ‡πœΌ)+𝑃𝑇12̇𝐺+π‘†ξ‚‡π‘ƒΜ‡πœΌ=0,(38)which is reduced to the Eular equation in Μ‡πœΌ:𝐺(π‘ž)̈𝜼+12̇𝐺+π‘†ξ‚‡Μ‡πœΌ=0,(39)or equivalentlyΜˆπœ‚π‘˜+Ξ“π‘˜π‘–π‘—Μ‡πœ‚π‘–Μ‡πœ‚π‘—=0,π‘˜=1,…,π‘›βˆ’1(=3),(40)where 𝐺(π‘ž)=𝑃𝑇𝐺(π‘ž)𝑃, Ξ“π‘˜π‘–π‘— the Christoffel symbol for (𝑔𝑖𝑗)=𝐺, and𝑆=π‘ƒπ‘‡π‘†π‘ƒβˆ’12̇𝑃𝑇𝐺𝑃+12𝑃𝑇𝐺̇𝑃,(41)which is skew symmetric, too. Note that the transformation 𝑄 is isometric, and (40) stands for the geodesic equation on the constraint Riemannian submanifold.

5. Hybrid Control for Redundant Systems and Stability on a Submanifold

Let us now reconsider a hybrid position/force feedback control scheme, which is of the form𝑒=𝑔(π‘ž)+πœ†π‘‘π½π‘‡(π‘ž)πœ•πœ‘πœ•π±+πΆΜ‡π‘ž+𝐽𝑇(π‘ž)ξ€·πœβˆšπ‘˜Μ‡π±+π‘˜Ξ”π±ξ€Έ,(42)where πœ‘(𝐱)=𝑧, ̇𝐱=(Μ‡π‘₯,̇𝑦,0)𝑇, Δ𝐱=(π‘₯βˆ’π‘₯𝑑,π‘¦βˆ’π‘¦π‘‘,0). This type of hybrid control is an extension of the McClamroch and Wang's method [7, 8] to cope with a robotic system that is subject to redundancy in DOFs. From now on, we redefine the Jacobian matrix 𝐽(π‘ž) as 2Γ—4 matrix 𝐽(π‘ž)=(πœ•π‘₯/πœ•π‘žπ‘‡,πœ•π‘¦/πœ•π‘žπ‘‡) because we consider a special case of πœ‘(𝐱)=𝑧, that is, the constraint π‘§βˆ’πœ‰=0, and solution trajectories π‘ž(𝑑) that satisfy 𝑧(π‘ž(𝑑))βˆ’πœ‰=0. In relation to this, denote also the two-dimensional vector (π‘₯,𝑦) by π‘₯ and (Ξ”π‘₯,Δ𝑦) by Δ𝐱. It should be noted that the orthogonality relationship among (π‘₯,𝑦,𝑧) coordinates in 𝐄3 does not imply the orthogonality among πœ•π‘₯/πœ•π‘ž, πœ•π‘¦/πœ•π‘ž, and πœ•π‘§/πœ•π‘ž(=πœ•πœ‘/πœ•π‘ž). Therefore, πœ•π‘§/πœ•π‘ž is not orthogonal to 𝐽𝑇(π‘ž)(=(πœ•π‘₯/πœ•π‘ž,πœ•π‘¦/πœ•π‘ž)), and hence the last term of the right hand side of (42) contains both components of image (𝑀) and ker(𝑀).

Now, substituting this into (27) yields𝐺(π‘ž)Μˆπ‘ž+12̇𝐺+𝑆+πΆξ‚‡Μ‡π‘ž+𝐽𝑇(π‘ž)ξ€½πœβˆšπ‘˜Μ‡π±+π‘˜Ξ”π±ξ€Ύ=βˆ’Ξ”πœ†πœ•π‘§(π‘ž)πœ•π‘ž.(43)Evidently the inner product of (43) and Μ‡π‘ž under the constraint 𝑧(π‘ž)=πœ‰ leads toddπ‘‘πΈξ€·π‘ž,Μ‡π‘žξ€Έ=βˆ’Μ‡π‘žπΆΜ‡π‘žβˆ’πœβˆšπ‘˜β€–β€–Μ‡π±β€–β€–2,(44)whereπΈξ€·π‘ž,Μ‡π‘žξ€Έ=12Μ‡π‘žπ‘‡πΊ(π‘ž)Μ‡π‘ž+π‘˜2‖Δ𝐱‖2.(45)Unfortunately, this quantity is not positive definite in the tangent bundle π‘‡πΉπœ‰. Nevertheless, it is possible to see that magnitudes of Μ‡π‘ž(𝑑) and Δ𝐱(𝑑) remain small if at initial time 𝑑=0 both magnitudes β€–Μ‡π‘ž(0)β€– and ‖Δ𝐱(0)β€– are set small. Next, let us introduce the equilibrium manifold defined by the setEM1=ξ€½π‘žβˆΆπ‘§(π‘ž)=πœ‰,π‘₯(π‘ž)=π‘₯𝑑,𝑦(π‘ž)=𝑦𝑑,(46)which is of one dimension. Fortunately as discussed in the paper [9], it is possible to confirm that a modified scalar functionπ‘Šπ›Ό=πΈξ€·π‘ž,Μ‡π‘žξ€Έ+π›ΌΜ‡π‘žπ‘‡πΊ(π‘ž)π‘ƒπœ‘π½π‘‡(π‘ž)Δ𝐱(47)becomes positive definite in π‘‡πΉπœ‰ with an appropriate positive parameter 𝛼>0, provided that the physical scale of the robot is ordinary, and feedback gains 𝐢 and π‘˜ are chosen adequately. Here, in (47), π‘ƒπœ‘ is defined as 𝐼4βˆ’π‘€π‘€π‘‡/‖𝑀‖2. Furthermore, if in a Riemannian ball in πΉπœ‰ defined as 𝐡(π‘ž(0),π‘Ÿ0)={π‘žβˆΆπ‘‘(π‘ž,π‘ž(0))<π‘Ÿ0}, the Jacobian matrix 𝐽(π‘ž) is nondegenerate, then it can be expected that there exist positive numbers πœŽπ‘š and πœŽπ‘€ such thatπœŽπ‘€πΌ2β‰₯𝐽𝐽𝑇β‰₯π½π‘ƒπœ‘π½π‘‡β‰₯πœŽπ‘šπΌ2.(48)Further, to simplify the argument, we choose 𝐢=𝑐𝐼4 with a constant 𝑐>0. Then, it follows from differentiation of 𝑉(=Μ‡π‘žπ‘‡πΊ(π‘ž)π‘ƒπœ‘π½π‘‡(π‘ž)Δ𝐱) thaṫ𝑉=βˆ’ξ€½πœβˆšπ‘˜Μ‡π±+π‘˜Ξ”π±ξ€Ύπ‘‡π½π‘ƒπœ‘π½π‘‡Ξ”π±βˆ’π‘Μ‡π‘žπ‘‡π‘ƒπœ‘π½π‘‡Ξ”π±+Μ‡π‘žπ‘‡πΊπ‘ƒπœ‘π½π‘‡Μ‡π±βˆ’β„Ž(Μ‡π‘ž,𝐺)Δ𝐱,(49)whereβ„Ž(Μ‡π‘ž,𝐺)=Μ‡π‘žπ‘‡ξ‚†ξ‚€12Μ‡πΊπ‘ƒπœ‘βˆ’π‘†π‘ƒπœ‘βˆ’πΊΜ‡π‘ƒπœ‘ξ‚π½π‘‡βˆ’πΊπ‘ƒπœ‘Μ‡π½π‘‡ξ‚‡.(50)This 1Γ—2-vector β„Ž is quadratic in components of Μ‡π‘ž, and each coefficient of Μ‡π‘žπ‘–Μ‡π‘žπ‘— (for 𝑖,𝑗=1,…,4) is at most of order of the maximum spectre radius of 𝐺(π‘ž) denoted by 𝑔𝑀. Hence, it follows from (49) that||β„Žξ€·Μ‡π‘ž,𝐺Δ𝐱||≀𝑔𝑀𝑙0β€–β€–Μ‡π‘žβ€–β€–2‖Δ𝐱‖,(51)where 𝑙0 signifies a constant that is of order of the maximum link length (because the Jacobian matrix πœ•π±/πœ•π‘žπ‘‡ is homogeneously related to the three link lengths of the robot shown in Figure 4). Further, note that Μ‡π‘žπ‘‡π‘ƒπœ‘=0 in this case and remark thatΜ‡π‘žπ‘‡πΊπ‘ƒπœ‘π½π‘‡Μ‡π±β‰€12Μ‡π‘žπ‘‡πΊπΊΜ‡π‘ž+12Μ‡π±π‘‡π½π‘ƒπœ‘π½π‘‡Μ‡π±,(52)βˆ’πœβˆšπ‘˜Μ‡π±π‘‡π½π‘ƒπœ‘π½π‘‡Ξ”π±β‰€πœπ‘˜2Ξ”π±π‘‡π½π‘ƒπœ‘π½π‘‡Ξ”π±+𝜁2̇𝐱𝑇𝐽𝐽𝑇̇𝐱.(53)Thus, substituting (51) to (52) into (49) yieldsΜ‡π‘‰β‰€βˆ’ξ‚†π‘˜βˆ’12βˆ’πœπ‘˜2ξ‚‡Ξ”π±π‘‡π½π‘ƒπœ‘π½π‘‡Ξ”π±+12Μ‡π‘žπ‘‡πΊπΊΜ‡π‘ž+1+𝜁2̇𝐱𝑇𝐽𝐽𝑇̇𝐱+𝑔𝑀𝑙0β€–β€–Μ‡π‘žβ€–β€–2‖Δ𝐱‖.(54)With the aid of a positive parameter 𝛼>0, it is easy to see thatπ‘Šπ›Όβ‰₯πΈβˆ’π›Ό2Μ‡π‘žπ‘‡πΊ(π‘ž)𝐺(π‘ž)Μ‡π‘žβˆ’π›Ό2Ξ”π±π‘‡π½π‘ƒπœ‘π½π‘‡Ξ”π±=12Μ‡π‘žπ‘‡{πΊβˆ’π›ΌπΊπΊ}Μ‡π‘ž+12Ξ”π±π‘‡ξ€·π‘˜πΌ2βˆ’π›Όπ½π‘ƒπœ‘π½π‘‡ξ€ΈΞ”π±.(55)

Now, suppose that the robot has an ordinary physical scale with𝑔𝑀≀0.001[kgm2],𝑙0≀0.2[m].(56)Then, it is possible to see thatΜ‡π‘Šπ›Ό=̇𝐸+π›ΌΜ‡π‘‰β‰€βˆ’π›Όπ‘˜πœŽπ‘š2‖Δ𝐱‖2βˆ’π‘3β€–β€–Μ‡π‘žβ€–β€–2,(57)provided that ‖Δ𝐱(𝑑)β€–<0.3[m], (1+πœπ‘˜)β‰€π‘˜, and πœβˆšπ‘˜>𝛼(1+𝜁)/2. From the practical point of view of control design for the handwriting robot, 𝜁 is set around 𝜁=√2/2, and π‘˜ is chosen between 5.0∼20.0[kg/s2] (see [9]). On account of (48) and (56),π‘Šπ›Όβ‰₯ξ€·1βˆ’π›Όπ›Ύ0𝐸,(58)and similarly,π‘Šπ›Όβ‰€ξ€·1+𝛼𝛾0𝐸,(59)where 𝛾0=max(𝑔𝑀,πœŽπ‘€/π‘˜). Then, if the damping factor can be selected to satisfy both inequalities 𝑐β‰₯(3/2)(π›ΌπœŽπ‘šπ‘”π‘€) and 𝑐>𝛼𝑔𝑀𝑙0, then (57) is reduced toΜ‡π‘Šπ›Όβ‰€βˆ’π›ΌπœŽπ‘š1+𝛼𝛾0π‘Šπ›Ό,(60)where 𝛾0 can be set as 𝛾0=πœŽπ‘€/π‘˜ which is larger than 𝑔𝑀. Hence, by choosing 𝛼=1/2𝛾0=π‘˜/2πœŽπ‘€, it follows from (60) and (59) thatπ‘Šπ›Ό(𝑑)=π‘Šπ›Ό(0)π‘’βˆ’π›Ύπ‘‘,(61)where 𝛾=π‘˜πœŽπ‘š/3πœŽπ‘€.

Now, suppose that π‘žβˆ— in EM1∩𝐡(π‘ž(0),π‘Ÿ0) is the minimizing point that connects with π‘ž(0) among all geodesics from π‘ž(0) to any point of EM1∩𝐡(π‘ž(0),π‘Ÿ0). We call π‘žβˆ— a reference point corresponding to π‘ž(0).

Definition 1 (stable Riemannian ball on a submanifold). If for any πœ€>0, there exists the number 𝛿(πœ€)>0 such that any solution trajectory (orbit) of (43) starting from an arbitrary initial position inside 𝐡(π‘žβˆ—,𝛿(πœ€)) with Μ‡π‘ž(0)=0 remains inside 𝐡(π‘žβˆ—,πœ€) for any 𝑑>0, then the reference point π‘žβˆ— on EM1 is said to be stable on a submanifold (see Figure 5).
It can be concluded from the exponential convergence of π‘Šπ›Ό to zero and the inequality 𝐸≀2π‘Šπ›Ό when 𝛼=1/2𝛾0 that any point inside 𝐡(π‘žβˆ—,𝛿(πœ€)) included in 𝐡(π‘ž(0),π‘Ÿ1) for some π‘Ÿ1(<π‘Ÿ0) is stable on a submanifold and further such a solution trajectory converges asymptotically to some π‘žβˆž on the equilibrium manifold EM1 in an exponential speed of convergence. This can be well understood as a natural extension of the well-known Dirichlet-Lagrange stability under holonomic constraints to a system with DOF-redundancy. The details of the proof are presented in Appendix .

It should be noticed from the proof in Appendix that asymptotic convergence of the solution trajectory to some π‘žβˆž on the equilibrium manifold implies also the asymptotic convergence of constraint force πœ†(𝑑) to πœ†π‘‘ as π‘‘β†’βˆž because Ξ”πœ†=πœ†βˆ’πœ†π‘‘ is expressed asΞ”πœ†(𝑑)=1π‘€π‘‡πΊβˆ’1π‘€ξ‚»ξ“π‘˜ξ‚΅Μ‡π‘€Μ‡π‘žπ‘˜βˆ’ξ“π‘–,π‘—π‘€π‘˜Ξ“π‘˜π‘–π‘—Μ‡π‘žπ‘–Μ‡π‘žπ‘—ξ‚Άβˆ’π‘€π‘‡π½π‘‡ξ€·πœβˆšπ‘˜Μ‡π±+π‘˜Ξ”π±ξ€Έξ‚Ό,(62)and this right hand side converges to zero as π‘‘β†’βˆž.

The stability notion of a Riemannian ball in a neighborhood of a reference equilibrium state π‘žβˆ— on EM1 is extended to cope with the case that the initial velocity vector Μ‡π‘ž(0) is not zero. To do this, we define an extended Riemannian ball in the tangent bundle π‘€Γ—π‘‡πΉπœ‰ around (π‘žβˆ—,Μ‡π‘ž=0) in such a way thatπ΅ξ€½ξ€·π‘žβˆ—,0ξ€Έ;ξ€·π‘Ÿ0,π‘ŸπΎξ€Έξ€Ύ=ξ‚†ξ€·π‘ž,Μ‡π‘žξ€ΈβˆΆπ‘‘ξ€·π‘ž,π‘žβˆ—ξ€Έ<π‘Ÿ0,(1/2)Μ‡π‘žπ‘‡πΊ(π‘ž)Μ‡π‘ž<π‘ŸπΎξ‚‡,(63)where 𝑑(π‘ž,π‘žβˆ—) denotes the distance between π‘ž and π‘žβˆ— restricted to the submanifold πΉπœ‰, and 𝐺 is defined below (40).

Definition 2 (asymptotic stability on a submanifold). If for any πœ€>0, there exist numbers 𝛿(πœ€) and 𝛿𝐾(πœ€) such that any solution trajectory of (43) starting from an arbitrary initial position and velocity inside 𝐡{(π‘žβˆ—,0);(𝛿(πœ€),𝛿𝐾(πœ€))} remains in 𝐡{(π‘žβˆ—,0);(πœ€,π‘ŸπΎ)} and further converges asymptotically to some equilibrium point π‘žβˆžβˆˆEM1 with still state, then the reference point with its posture on EM1 is said to be asymptotically stable on a constraint submanifold.

It should be remarked that Bloch et al. [10] introduces originally the concept of stabilization for a class of nonholonomic dynamic systems based upon a certain configuration space. The redefinition of stability concepts introduced above is free from any choice of configuration spaces (local coordinates) and assumptions on an invertibility condition (that is almost equivalent to nonlinear control based on compensation for nonlinear inertia-originated terms). Liu and Li [11] also gave a geometric approach to modeling of constrained mechanical systems based upon orthogonal projection maps without deriving a compact explicit form of the Euler equation like (40) with a reduced dimension due to constraints. Therefore, the proposed control scheme was developed on the basis of compensation for the inertia-originated nonlinear terms (that is almost equivalent to the computed torque method). A naive idea of stability on a manifold by using different metrics for the constrained submanifold and its tangent space was first presented in [9] and used in stabilization control of robotic systems with DOF redundancy [12, 13].

6. 2-dimensional Stable Grasp of a Rigid Object with Arbitrary Shape

Consider a control problem for stable grasping of a 2D rigid object by a pair of planar multijoint robot fingers with hemispherical fingertips as shown in Figure 6. In this figure, the two robots are installed on the horizontal π‘₯𝑦-plane 𝐄2. We denote the object mass center by π‘‚π‘š with the coordinates (π‘₯π‘š,π‘¦π‘š) expressed in the inertial frame. On the other hand, we express a local coordinate system fixed at the object by π‘‚π‘š-π‘‹π‘Œ together with unit vectors 𝐫𝑋 and π«π‘Œ along the 𝑋-axis and π‘Œ-axis, respectively (see Figure 7). The left-hand side surface of the object is expressed by a curve 𝑐(𝑠) with local coordinates (𝑋(𝑠),π‘Œ(𝑠)) in terms of arc length parameter 𝑠 as shown in Figure 7.

First, suppose that at the left-hand contact point 𝑃1 the fingertip of the left finger is contacting with the object. Denote the unit normal at 𝑃1 by 𝐧1 and the unit tangent vector by 𝐞1. Note that 𝐧1 is normal to both the object surface and finger-end sphere at 𝑃1, and 𝐞1 is tangent to them at 𝑃1, too. If we denote position 𝑃1 by local coordinates (𝑋(𝑠),π‘Œ(𝑠)) fixed at the object (see Figure 7), then the angle from the 𝑋-axis to the unit normal 𝐧1 is assumed to be determined by a function on the curve:πœƒ1(𝑠)=arctξ‚΅π‘‹ξ…ž(𝑠)π‘Œξ…ž(𝑠)ξ‚Ά,(64)where π‘‹ξ…ž(𝑠)=d𝑋(𝑠)/d𝑠 and π‘Œβ€²=dπ‘Œ(𝑠)/d𝑠. In this paper, all angles are set positive in counterclockwise direction. Then,𝑃1π‘ƒξ…ž1=𝑙𝑛1(𝑠)=βˆ’π‘‹(𝑠)cosπœƒ1(𝑠)+π‘Œ(𝑠)sinπœƒ1(𝑠),(65)which is dependent only on 𝑠 and, therefore, a shape function of the object. On the other hand, this length can be expressed by using the inertial frame coordinates in the following way (see Figure 8):𝑃1π‘ƒξ…ž1=ξ€·π‘₯π‘šβˆ’π‘₯01ξ€Έcosξ€·πœƒ+πœƒ1(𝑠)ξ€Έβˆ’ξ€·π‘¦π‘šβˆ’π‘¦01ξ€Έsinξ€·πœƒ+πœƒ1(𝑠)ξ€Έβˆ’π‘Ÿ1.(66)Hence, the left-hand contact constraint can be expressed by the holonomic constraint𝑄1=βˆ’ξ€·π‘₯π‘šβˆ’π‘₯01ξ€Έcosξ€·πœƒ+πœƒ1ξ€Έ+ξ€·π‘¦π‘šβˆ’π‘¦01ξ€Έsinξ€·πœƒ+πœƒ1ξ€Έ+ξ€·π‘Ÿ1+𝑙𝑛1(𝑠)ξ€Έ=0,(67)where 𝑙𝑛1(𝑠) denotes the right-hand side of (65) and πœƒ1=πœƒ1(𝑠) for abbreviation.

Next, we note that the length π‘‚π‘šπ‘ƒξ…ž1 can be regarded also as a shape function of the object given byπ‘‚π‘šπ‘ƒξ…ž1=𝑙𝑒1(𝑠)=𝑋(𝑠)sinπœƒ1+π‘Œ(𝑠)cosπœƒ1.(68)On the other hand, this quantity can be also expressed asπ‘‚π‘šπ‘ƒξ…ž1=π‘Œ1(𝑑)=βˆ’ξ€·π‘₯π‘šβˆ’π‘₯01ξ€Έsinξ€·πœƒ+πœƒ1ξ€Έβˆ’ξ€·π‘¦π‘šβˆ’π‘¦01ξ€Έcosξ€·πœƒ+πœƒ1ξ€Έ.(69)As discussed in [13], in the light of the book [14], pure rolling at 𝑃1 is defined by the condition that the translational velocity of 𝑃1 on the finger sphere is equal to that of 𝑃1 on the object contour along the 𝐞1-axis. Hence, by denoting 𝑝1=π‘ž11+π‘ž12+π‘ž13, we haveβˆ’π‘Ÿ1̇𝑝1+π‘Ÿ1Μ‡πœƒ=βˆ’Μ‡π‘Œ1=ξ€·Μ‡π‘₯π‘šβˆ’Μ‡π‘₯01ξ€Έsinξ€·πœƒ+πœƒ1ξ€Έ+ξ€·Μ‡π‘¦π‘šβˆ’Μ‡π‘¦01ξ€Έcosξ€·πœƒ+πœƒ1ξ€Έ+ξ€½ξ€·π‘₯π‘šβˆ’π‘₯01ξ€Έcosξ€·πœƒ+πœƒ1ξ€Έβˆ’ξ€·π‘¦π‘šβˆ’π‘¦01ξ€Έsinξ€·πœƒ+πœƒ1)ξ€ΎΜ‡πœƒ,(70)which from (67) can be reduced toβˆ’π‘Ÿ1̇𝑝1βˆ’π‘™π‘›1Μ‡πœƒ+ξ€·Μ‡π‘₯01βˆ’Μ‡π‘₯ξ€Έsinξ€·πœƒ+πœƒ1ξ€Έ+̇𝑦01βˆ’Μ‡π‘¦ξ€Έcosξ€·πœƒ+πœƒ1ξ€Έ=0.(71)This constraint form is Pfaffian. As to the contact point 𝑃2 at the right-hand finger end, a similar nonholonomic constraint can be obtained. Thus, by introducing Lagrange's multipliers 𝑓1 and 𝑓2 associated with holonomic constraints 𝑄𝑖=0(𝑖=1,2), it is possible to construct a Lagrangian:𝐿=12ξ“Μ‡π‘žπ‘‡π‘–πΊπ‘–ξ€·π‘žπ‘–ξ€ΈΜ‡π‘žπ‘–+12𝑀̇π‘₯2+𝑀̇𝑦2+πΌΜ‡πœƒ2ξ€Ύβˆ’π‘“1𝑄1βˆ’π‘“2𝑄2,(72)where π‘žπ‘– denote the joint vector for finger 𝑖, 𝐺𝑖(π‘žπ‘–) the inertia matrix for finger 𝑖, 𝑀 and 𝐼 denote the mass and inertia moment of the object. Since both the rolling constraints are Pfaffian, it is possible to associate (71) and its corresponding form at 𝑃2 with another multipliers πœ†π‘–(𝑖=1,2) and regard them as external forces. Thus, by applying the variational principle to the Lagrangian together with the external forces, we obtain the Lagrange equation of motion of the overall fingers-object system:πΌΜˆπœƒβˆ’π‘“1π‘Œ1+𝑓2π‘Œ2βˆ’πœ†1𝑙𝑛1+πœ†2𝑙𝑛2=0,(73)π‘€ξ‚΅Μˆπ‘₯Μˆπ‘¦ξ‚Άβˆ’π‘“1𝐧1βˆ’π‘“2𝐧2βˆ’πœ†1𝐞1βˆ’πœ†2𝐞2=0,(74)πΊπ‘–ξ€·π‘žπ‘–ξ€ΈΜˆπ‘žπ‘–+12̇𝐺𝑖+π‘†π‘–ξ‚‡Μ‡π‘žπ‘–+π‘“π‘–π½π‘‡π‘–ξ€·π‘žπ‘–ξ€Έπ§π‘–+πœ†π‘–ξ€½π½π‘‡π‘–ξ€·π‘žπ‘–ξ€Έπžπ‘–