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

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 “submersion” 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 𝐑^𝑛 [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 “law 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 𝑚(=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 “law 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 “blind 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(𝑞)̈𝑞+12𝐻(𝑞)+𝑆(𝑞,̇𝑞)̇̇𝑞+𝑔(𝑞)=𝑢,(1)where 𝑞=(𝑞₁,𝑞₂)^𝑇 denotes the vector of joint angles, 𝐻(𝑞) denotesthe inertia matrix, 𝑆(𝑞,𝑞̇)𝑞̇ the gyroscopic force term including centrifugal and Coriolis forces, 𝑔(𝑞) the gradient vector of a potential function 𝑃(𝑞) due to the gravity with respect to 𝑞, and 𝑢 the joint torque generated by joint actuators [4]. It is well known that the inertia matrix 𝐻(𝑞) is symmetric and positive definite, and there exist a positive constant ℎ_𝑚 together with a positive definite constant diagonal matrix 𝐻₀ such thatℎ_𝑚𝐻₀≤𝐻(𝑞)≤𝐻₀(2)for any 𝑞. It should be also noted that 𝑆(𝑞,𝑞̇) is skew symmetric and linear and homogeneous in 𝑞̇. More in detail, the 𝑖𝑗th entry of 𝑆(𝑞,𝑞̇) denoted by 𝑠_𝑖𝑗 can be described in the form [3]𝑠_𝑖𝑗=12𝜕𝑞𝜕_𝑗^𝑛_𝑘=1𝑘̇𝑞ℎ_𝑖𝑘−𝜕𝑞𝜕_𝑖^𝑛_𝑘=1𝑘̇𝑞ℎ_𝑗𝑘,(3)where 𝐻(𝑞)=(ℎ_𝑖𝑗(𝑞)), from which it follows apparently that 𝑠_𝑖𝑗=−𝑠_𝑗𝑖. 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(𝑞) is a constant or a sinusoidal function of components of joint angle vector 𝑞. That is, each element of 𝐻(𝑞) and 𝑔(𝑞) is differentiable of class 𝐶^∞ (infinitely differentiable in 𝑞).

Figure 1 

A 2-DOF part of vertically revolute robot PUMA-560.

When two joint angles 𝜃₁ and 𝜃₂ are given in 𝜃_𝑖∈(−𝜋,𝜋], 𝑖=1,2, for the 2 DOF robot arm shown in Figure 1, the posture 𝑝(𝜃₁,𝜃₂) 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 𝑝∈𝑀 with joint angles 𝑝=(𝜃₁,𝜃₂) and any number 𝛼>0, a set of all 𝑝′=(𝜃^1,𝜃^2) is defined as𝑈_𝑝,𝛼=𝑝^∶‖‖𝑝^−𝑝‖‖_𝐻<𝛼,(4)where‖‖𝑝^−𝑝‖‖_𝐻=_𝑖,𝑗ℎ_𝑖𝑗(𝑝)𝜃^𝑖−𝜃_𝑖𝜃^𝑗−𝜃_𝑗(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 𝑈 that is homeomorphic to an open subset Ω of 2-dimensional numerical space 𝐑². Such a homeomorphism 𝜙∶𝑈→Ω is called a coordinate chart. In fact, a neighborhood 𝑈_𝑝,𝛼 of posture 𝑝 with joint angles (𝜃₁,𝜃₂) in Figure 1 can be mapped to an open set Ω in 𝐑² with 2 dimensional numerical coordinates (𝑞¹,𝑞²) 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 𝑞₁ and 𝑞₂ 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 (𝑞¹,…,𝑞^𝑛) can be identically chosen as a set of 𝑛 independent joint angles (𝜃₁,…,𝜃_𝑛) by setting 𝑞^𝑖=𝜃_𝑖(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=(𝑥,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 𝐶^∞.

Figure 2 

A homeomorphism $\varphi :U\to \mathrm{\Omega }$ is called a chart.

Figure 3 

A two-dimensional torus is expressed by $T^2=S^1\times S^1$.

Now, it is necessary to define a tangent vector to an abstract differentiable manifold 𝑀 at 𝑝∈𝑀. Let 𝐼 be an interval (−𝜖,𝜖) and define a curve 𝑐(𝑡) by a mapping 𝑐∶𝐼→𝑀 such that 𝑐(0)=𝑝. A tangent vector to 𝑀 at 𝑝 is an equivalence class of curves 𝑐∶𝐼→𝑀 for the equivalence relation ∼ defined by

𝑐∼𝑐if and only if, in a coordinate chart (𝑈,𝜙) around 𝑝, we have (𝜙∘𝑐)^(0)=(𝜙∘𝑐)^(0), where symbol ()′ means differentiation of () with respect to 𝑡∈𝐼. 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 𝑇_𝑝𝑀 and call it the tangent space at 𝑝∈𝑀. It has an 𝑛-dimensional linear space structure like 𝐑^𝑛. 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 𝑔_𝑝∶𝑇_𝑝𝑀×𝑇_𝑝𝑀→𝐑 such that 𝑝→𝑔_𝑝 is of class 𝐶^∞ and 𝑔_𝑝(𝑢,𝑣) for 𝑢∈𝑇_𝑝𝑀 and 𝑣∈𝑇_𝑝𝑀 is a symmetric positive definite quadratic form𝑔_𝑝(𝑢,𝑣)=^𝑛_𝑖_,𝑗=1𝑔_𝑖𝑗(𝑝)𝑢^𝑖𝑣^𝑗.(6)

Suppose that 𝑀 is a connected Riemannian manifold. If 𝑐∶𝐼[𝑎,𝑏]→𝑀 is a curve segment of class 𝐶^∞, we define the length of 𝑐 to be𝐿(𝑐)==^𝑏𝑎‖‖𝑐̇(𝑡)‖‖d𝑡_𝑏𝑎𝑔_𝑐(𝑡)𝑐̇(𝑡),𝑐̇(𝑡)d𝑡,(7)where we assume 𝑐̇(𝑡)≠0 for any 𝑡∈𝐼 and call such a curve segment to be regular. A mapping of class 𝐶^∞𝑐∶[𝑎,𝑏]→𝑀 is called a piecewise regular curve segment (for brevity, we call it an admissible curve) if there exists a finite subdivision 𝑎=𝑎₀<𝑎₁<⋯<𝑎_𝑘=𝑏 such that 𝑐(𝑡) for 𝑡∈[𝑎_𝑖−1,𝑎_𝑖] is a regular curve for 𝑖=1,…,𝑘. Then, it is possible to define for any pair of points 𝑝,𝑝′∈𝑀 the Riemannian distance 𝑑(𝑝,𝑝^) to be the infimum of all admissible curves from 𝑝 to 𝑝^. 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(𝑐̃) for any other admissible curve 𝑐̃ 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 𝑝^ there exists at least a minimizing curve 𝑐(𝑡), 𝑡∈[𝑎,𝑏], with 𝑐(𝑎)=𝑝 and 𝑐(𝑏)=𝑝^. If such a minimizing curve 𝑐(𝑡) is described with the aid of coordinate chart (𝑈,𝜙) as 𝜙(𝑐(𝑡))=(𝑞¹(𝑡),…,𝑞^𝑛(𝑡)), then 𝑞(𝑡)=𝜙(𝑐(𝑡)) satisfies the 2nd-order differential equationd²d𝑡²𝑞^𝑘(𝑡)+^𝑛_𝑖_,𝑗=1Γ^𝑘𝑖𝑗(𝑐(𝑡))d𝑞^𝑖(𝑡)d𝑡d𝑞^𝑗(𝑡)(𝑘=d1𝑡,…=,𝑛0),,(8)where Γ^𝑘𝑖𝑗 denotes Christoffel's symbol defined byΓ^𝑘𝑖𝑗=12^𝑛_ℎ=1𝑔^𝑘ℎ𝜕𝜕𝑔𝑞^𝑖ℎ𝑗+𝜕𝑔^𝑗ℎ𝜕𝑞^𝑖−𝜕𝑔^𝑖𝑗𝜕𝑞^ℎ,(9)and (𝑔^𝑘ℎ) denotes the inverse of matrix (𝑔_𝑘ℎ). A curve 𝑞(𝑡)∶𝐼→𝑈 satisfying (8) together with 𝜙⁻¹(𝑞(𝑡)) is called a geodesic, and (8) itself is called the Euler-Lagrange equation or the geodesic equation.

Given a 𝐶^∞-class curve 𝑐(𝑡)=𝐼[𝑎,𝑏]→𝑀, the quantity𝐸(𝑐)=12^𝑏𝑎=1‖‖𝑐̇(𝑡)‖‖²d𝑡2^𝑏𝑎𝑔_𝑐(𝑡)𝑐̇(𝑡),𝑐̇(𝑡)d𝑡(10)is called the energy of the curve. Then, by applying the Cauchy-Schwartz inequality for (7), we have𝐿(𝑐)²≤2(𝑏−𝑎)𝐸(𝑐).(11)Further, the equality of (11) follows if and only if ‖𝑐̇(𝑡)‖ is constant. It is also possible to see that if 𝑐(𝑡) is a geodesic with 𝑐(𝑎)=𝑝 and 𝑐(𝑏)=𝑝^, then for any other curve 𝑐(𝑡) with the same endpoints, it holds𝐸(𝑐)=𝐿(𝑐)²2(𝑏−𝑎)≤𝐿𝑐²2(𝑏−𝑎)≤𝐸𝑐.(12)The equalities hold if and only if 𝑐(𝑡) is also a geodesic. Conversely, if 𝑐(𝑡) with 𝑐(𝑎)=𝑝 and 𝑐(𝑏)=𝑝^ is a 𝐶^∞ curve that minimizes the energy and makes 𝑔_𝑐(𝑡)(𝑐̇(𝑡),𝑐̇(𝑡)) constant, then 𝑐(𝑡) becomes a geodesic connecting 𝑐(𝑎)=𝑝 and 𝑐(𝑏)=𝑝^. In mechanics, 𝐸(𝑐) is usually called “action of 𝑐,” and 𝑐(𝑡) 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 𝑥𝑦-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 form𝐻(𝑞)𝑞̈+12𝐻̇(𝑞)+𝑆𝑞,𝑞̇𝑞̇=𝑢,(13)where 𝑢 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(𝑞) (included in (1)) can be compensated by joint actuators through control input 𝑢. In general, we can represent a posture 𝑝 of the robot as a physical entity by a family of joint angles 𝜃_𝑖(𝑖=1,…,𝑛), which can be expressed by a point Θ=(𝜃₁,…,𝜃_𝑛) in the 𝑛-dimensional numerical space 𝐑^𝑛. In fact, we can naturally imagine an isomorphism 𝜙∶𝑈→Ω, where 𝑈⊂𝑀, and Ω is an open subset of 𝐑^𝑛. In other words, a local coordinate chart 𝜙(𝑈)(=Ω) can be treated to be identical to 𝑈 itself, an open subset of 𝑀, by regarding 𝑞=(𝑞¹,…,𝑞^𝑛)^𝑇 (“𝑇” denotes transpose and hence 𝑞 a column vector) identical to Θ by setting 𝑞^𝑖=𝜃_𝑖(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 𝑇^𝑛 as an 𝑛-tuple direct product of 𝑆¹∶𝑇^𝑛=𝑆¹×⋯×𝑆¹. Hence, a robot posture 𝑝∈𝑀 can be represented by a point Θ on 𝑇^𝑛 and also expressed by a joint vector 𝑞 in 𝐑^𝑛.

From the definition of inertia matrices, 𝐻(𝑞) is symmetric and positive definite, and the kinetic energy is expressed as a quadratic form𝐾𝑞,𝑞̇=12𝑞̇^𝑇𝐻(𝑞)𝑞̇.(14)Hence, the equation of motion of the robot is expressed by Lagrange's equationdd𝑡𝜕𝜕𝑞̇𝐿𝑞,𝑞̇−𝜕𝜕𝑞𝐿𝑞,𝑞̇=𝑢,(15)where 𝐿(𝑞,𝑞̇)=𝐾(𝑞,𝑞̇), and 𝑢 stands for a generalized external force vector. It is interesting to note that in differential geometry, (15) can be described as_𝑖ℎ_𝑘𝑖𝑞̈^𝑖+_𝑖,𝑗Γ_𝑖𝑘𝑗(𝑞)𝑞̇^𝑖𝑞̇^𝑗=𝑢^𝑘,(16)where Γ_𝑖𝑘𝑗 denotes Christoffel's symbol of the first kind defined byΓ_𝑖𝑘𝑗=12𝜕𝜕ℎ𝑞^𝑗𝑘𝑖+𝜕ℎ^𝑖𝑘𝜕𝑞^𝑗−𝜕ℎ^𝑖𝑗𝜕𝑞^𝑘.(17)For later use, we introduce another Christoffel's symbol called the second kind as shown in the formulaΓ^𝑘𝑖𝑗=12^𝑛_𝑙=1ℎ^𝑙𝑘𝜕𝜕ℎ𝑞^𝑗𝑙𝑖+𝜕ℎ^𝑖𝑙𝜕𝑞^𝑗−𝜕ℎ^𝑖𝑗=1𝜕𝑞^𝑙2^𝑛_𝑙=1ℎ^𝑙𝑘Γ_𝑖𝑙𝑗,(18)where (ℎ^𝑙𝑘) denotes the inverse of (ℎ_𝑙𝑘), the inertia matrix 𝐻(𝑞)=(ℎ_𝑙𝑘). Since (ℎ_𝑘𝑙) and (ℎ^𝑘𝑙) are symmetric, it follows that Γ_𝑖𝑘𝑗=Γ_𝑗𝑘𝑖 and Γ^𝑘𝑖𝑗=Γ^𝑘𝑗𝑖. Now, we show that (13) is equivalent to (16) by bearing in mind that 𝐻̇(𝑞)=∑_𝑖{𝜕𝐻(𝑞)/𝜕𝑞^𝑖}𝑞̇^𝑖, and the skew symmetric matrix 𝑆(𝑞,𝑞̇) 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) that^𝑛_𝑗=1𝑠_𝑘𝑗𝑞̇^𝑗=^𝑛_𝑗=112𝜕𝜕𝑞^𝑗^𝑛_𝑖=1𝑞̇^𝑖ℎ_𝑘𝑖𝑞̇^𝑖−𝜕𝜕𝑞^𝑘^𝑛_𝑖=1𝑞̇^𝑖ℎ_𝑖𝑗=_𝑛𝑞̇^𝑗_𝑛𝑗=1_𝑖=112𝜕𝜕ℎ𝑞^𝑖𝑘𝑗−𝜕ℎ^𝑖𝑗𝜕𝑞^𝑘𝑞̇^𝑖𝑞̇^𝑗.(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 𝐻⁻¹(𝑞) yields𝑞̈^𝑘+_𝑖,𝑗Γ^𝑘𝑖𝑗𝑞̇^𝑖𝑞̇^𝑗=_𝑗ℎ^𝑘𝑗𝑢^𝑗,𝑘=1,…,𝑛.(20)If 𝑢=0, this expression is nothing, but the Euler-Lagrange equation shown in (8). By this reason, from now on, we use symbol 𝑔_𝑖𝑗(𝑞) instead of ℎ_𝑖𝑗(𝑞) for the inertia matrix 𝐻(𝑞) 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 𝑇_𝑝𝑀:⟨𝑣,𝑤⟩=𝑔_𝑖𝑗(𝑝)𝑣^𝑖𝑤^𝑗,(21)where 𝑣=𝑣^𝑖(𝜕/𝜕𝑞^𝑖)∈𝑇_𝑝𝑀 and 𝑤=𝑤^𝑗(𝜕/𝜕𝑞^𝑗)∈𝑇_𝑝𝑀, and the summation symbol ∑ in 𝑖 and 𝑗 is omitted, and 𝑞=(𝑞¹,…,𝑞^𝑛) 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,𝑞∈𝑀 can be joined by a geodesic of length 𝑑(𝑝,𝑞), that is, a curve satisfying (8) with shortest length, where𝑑(𝑝,𝑞)==iinnff^𝑏𝑎_𝑏𝑎‖‖𝑐̇(𝑡)‖‖d𝑡⟨𝑐̇(𝑡),𝑐̇(𝑡)⟩d𝑡(22)with 𝑐(𝑎)=𝑝 and 𝑐(𝑏)=𝑞.

As discussed in the previous section, geodesics are the critical points of the energy functional 𝐸(𝑐). Further, a geodesic curve 𝑐(𝑡) satisfies ‖𝑐̇(𝑡)‖=const. In fact, by regarding 𝑐(𝑡)=𝑞(𝑡) that is an orbit on Ω, we havedd𝑡⟨𝑞̇,𝑞̇⟩=dd𝑡𝑔^𝑖𝑗𝑞(𝑡)𝑞̇^𝑖=d(𝑡)𝑞̇^𝑗(𝑡)d𝑡𝑞̇^𝑇=𝑞̈_𝑇(𝑡)𝐺(𝑞(𝑡))𝑞̇(𝑡)=0,(𝑡)𝐺(𝑞)𝑞̇(𝑡)+𝑞̇^𝑇(𝑡)𝐺̇(𝑞)𝑞̇(𝑡)+𝑞̇^𝑇(𝑡)𝐺(𝑞)𝑞̈(𝑡)(23)where the equivalent expression𝐺(𝑞)𝑞̈+12𝐺̇(𝑞)+𝑆𝑞,𝑞̇𝑞̇=0(24)to (13) with 𝑢=0 is used, 𝐺(𝑞)=(𝑔_𝑖𝑗(𝑞)), and 𝑠_𝑖𝑗 of 𝑆 is given in the form of (3) (where ℎ_𝑖𝑗=𝑔_𝑖𝑗). It is also important to note that, on a local coordinate chart Ω⊂𝐑^𝑛 corresponding to a neighborhood 𝑈 of 𝑝∈𝑀, an orbit 𝑞(𝑡) parameterized by time 𝑡∈[𝑎,𝑏] and expressed by a solution to (20) (where 𝑢^𝑗 is of 𝐶^∞ in 𝑡) should satisfy^𝑛_𝑗=1̇𝑞^𝑗(𝑡)𝑢^𝑗(𝑡)=dd𝑡𝐸(𝑞(𝑡)),(25)or^𝑏𝑎𝑛_𝑗=1̇𝑞^𝑗(𝑡)𝑢^𝑗(𝑡)d𝑡=𝐸(𝑞(𝑏))−𝐸(𝑞(𝑎)),(26)as long as 𝑞(𝑡)∈Ω, where 𝐸(𝑞(𝑡))=(1/2)⟨̇𝑞(𝑡),̇𝑞(𝑡)⟩. When 𝑢(𝑡)=0, 𝐸(𝑞(𝑡))=const. and then the curve connecting 𝑝=𝜙⁻¹(𝑞(𝑎)) and 𝑝^=𝜙⁻¹(𝑞(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 𝑇^𝑛 is one of such choice of local coordinates corresponding to the choice of joint angles 𝑞=(𝑞¹,…,𝑞^𝑛)^𝑇. At the same time, it is important to note that, in differential geometry, choice of coordinates in the tangent space 𝑇_𝑝𝑀 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 𝑞=(𝑞¹,…,𝑞^𝑛)^𝑇, then the coordinates in the tangent space 𝑇_𝑝𝑀 should be chosen as the vector of joint angular velocities 𝜕/𝜕𝑞=(𝜕/𝜕𝑞¹,…,𝜕/𝜕𝑞^𝑛) correspondingly to ̇𝑞=(̇𝑞₁,…,̇𝑞_𝑛)^𝑇, from which the Riemannian metric 𝑔_𝑖𝑗(𝑞) 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 𝜑(𝐱)=𝜉, where 𝐱=(𝑥,𝑦,𝑧)^𝑇. It is well known that the Lagrange equation of motion of the system is written as𝐺(𝑞)̈𝑞+12𝐺+𝑆̇̇𝑞+𝑔(𝑞)=−𝜆𝜕𝜑(𝐱(𝑞))𝜕𝑞+𝑢,(27)where 𝜕𝜑/𝜕𝑞 can be decomposed into 𝜕𝜑(𝑞)/𝜕𝑞=𝐽^𝑇(𝑞)𝜕𝜑/𝜕𝐱 and 𝐽(𝑞)=𝜕𝐱/𝜕𝑞^𝑇. On the constraint manifold 𝐹_𝜉={𝑝∶𝑝∈𝑀and𝜑(𝐱(𝑝))=𝜉}, let us consider a smooth curve 𝑐(𝑡)∶𝐼[𝑎,𝑏]→𝐹_𝜉 that connects the given two points 𝑐(𝑎)=𝑝 and 𝑐(𝑏)=𝑝^, where 𝑝 and 𝑝^ belong to 𝐹_𝜉. The length of such a curve constrained to 𝐹_𝜉 is defined as𝐿(𝑐)=^𝑏𝑎𝑔_𝑖𝑗(𝑐(𝑡))̇𝑐^𝑖(𝑡)̇𝑐^𝑗(𝑡)d𝑡,(28)and consider the minimization𝑑𝑝,𝑝^=inf_{𝑐∈𝐹𝜉}𝐿(𝑐)(29)that should be called the distance between 𝑝 and 𝑝^ on the constraint manifold. Then, the minimizing curve called the geodesic denoted identically by 𝑞(𝑡)(=𝑐(𝑡)) must satisfy the Euler equation̈𝑞^𝑘(𝑡)+Γ^𝑘𝑖𝑗̇𝑞^𝑖(𝑡)̇𝑞^𝑗(𝑡)=−𝜆(𝑡)⋅grad𝜑(𝐱(𝑡))^𝑘,(30)together with the constraint condition 𝜑(𝐱(𝑡))=𝜉, wheregrad𝜑(𝐱(𝑡))=𝐺⁻¹(𝑞(𝑡))𝐽^𝑇(𝑞)𝜕𝜑𝜕𝐱,(31)and 𝐽^𝑇(𝑞)=𝜕𝐱^𝑇/𝜕𝑞. It should be noted that, from the inner product of (30) and 𝑤=𝐽^𝑇𝜕𝜑/𝜕𝐱, it follows that_𝑘𝑤^𝑘̈𝑞^𝑘+𝑤^𝑘Γ^𝑘𝑖𝑗̇𝑞^𝑖̇𝑞^𝑗=−𝜆_𝑘𝑤^𝑘𝑔^𝑘𝑗𝑤^𝑗.(32)Since the holonomic constraint 𝜑(𝐱(𝑞))=𝜉 implies ⟨𝑤,̇𝑞⟩=0, it follows that⁴_𝑘=1𝑤^𝑘̈𝑞^𝑘+dd𝑡𝑤^𝑘̇𝑞^𝑘=0.(33)Substituting this into (32), we obtain𝜆(𝑡)=1𝑤^𝑇𝐺⁻¹𝑤_𝑘𝑤̇^𝑘̇𝑞^𝑘−_𝑖,𝑗𝑤^𝑘Γ^𝑘𝑖𝑗̇𝑞^𝑖̇𝑞^𝑗.(34)From the Riemannian geometry, the constraint force 𝜆(𝑡) with the grad{𝜑(𝐱(𝑞))} stands for a component of the image space of 𝑤(=𝐽^𝑇(𝑞)𝜕𝜑/𝜕𝐱) that is orthogonal to the kernel 𝑇𝐹_𝜉 of 𝑤. 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, 𝜆(𝑡) should be regarded as a magnitude of the constraint force that presses the surface 𝜑(𝐱(𝑞))=𝜉 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 torques𝑢=𝜆_𝑑⋅𝐽^𝑇(𝑞)𝜕𝜑𝜕𝐱+𝑔(𝑞).(35)Then, by substituting this into (27), we obtain the Lagrange equation of motion under the constraint 𝜑(𝐱(𝑞))=𝜉:𝐺(𝑞)̈𝑞+12𝐺+𝑆̇̇𝑞=−𝜆−𝜆^𝑑𝐽^𝑇(𝑞)𝜕𝜑=−Δ𝜆𝐽_𝑇𝜕𝐱(𝑞)𝜕𝜑𝜕𝐱,(36)where Δ𝜆=𝜆−𝜆_𝑑. 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 𝜆(𝑡)(=𝜆_𝑑+Δ𝜆(𝑡)) positive. In a mathematical sense, exertion of the joint torque 𝜆_𝑑𝐽^𝑇(𝜕𝜑/𝐱) plays a role of “pressing” or “lifting” 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 Δ𝜆. 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 transformatioṅ𝑞=𝑃,𝑤‖𝑤‖⁻¹𝜼̇̇𝑧=𝑄𝑞,(37)̇where 𝑃 is a 4×3 matrix whose column vectors with the unit norm are orthogonal to 𝑤, and 𝜼̇ denotes a 3×1 matrix (3-dim. vector) and ̇𝑧 a scalar. Since 𝑄 is an orthogonal matrix, 𝑄⁻¹=𝑄^𝑇. Hence, if ̇𝑞∈ker(𝑤)(=𝑇𝐹_𝜉), then ̇𝑧=0. Restriction of (36) to the kernel space of 𝑤 can be attained by multiplying (36) by 𝑃^𝑇 from the left such that𝑃^𝑇𝐺(𝑞)dd𝑡(𝑃𝜼)+𝑃̇^𝑇12𝐺+𝑆̇𝑃𝜼=0,(38)̇which is reduced to the Eular equation in 𝜼̇:𝐺(𝑞)𝜼+̈12̇𝐺+𝑆𝜼=0,(39)̇or equivalentlÿ𝜂^𝑘+Γ^𝑘𝑖𝑗̇𝜂^𝑖̇𝜂^𝑗=0,𝑘=1,…,𝑛−1(=3),(40)where 𝐺(𝑞)=𝑃^𝑇𝐺(𝑞)𝑃, Γ^𝑘𝑖𝑗 the Christoffel symbol for (𝑔_𝑖𝑗)=𝐺, and𝑆=𝑃^𝑇𝑆𝑃−12𝑃̇^𝑇𝐺𝑃+12𝑃^𝑇𝐺𝑃,(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 form𝑢=𝑔(𝑞)+𝜆_𝑑𝐽^𝑇(𝑞)𝜕𝜑𝜕𝐱+𝐶̇𝑞+𝐽^𝑇(𝑞)𝜁√𝑘𝐱+𝑘Δ𝐱̇,(42)where 𝜑(𝐱)=𝑧, 𝐱=(̇𝑥,̇𝑦,0)̇^𝑇, Δ𝐱=(𝑥−𝑥_𝑑,𝑦−𝑦_𝑑,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(𝑞) as 2×4 matrix 𝐽(𝑞)=(𝜕𝑥/𝜕𝑞^𝑇,𝜕𝑦/𝜕𝑞^𝑇) because we consider a special case of 𝜑(𝐱)=𝑧, that is, the constraint 𝑧−𝜉=0, and solution trajectories 𝑞(𝑡) that satisfy 𝑧(𝑞(𝑡))−𝜉=0. In relation to this, denote also the two-dimensional vector (𝑥,𝑦) by 𝑥 and (Δ𝑥,Δ𝑦) by Δ𝐱. It should be noted that the orthogonality relationship among (𝑥,𝑦,𝑧) coordinates in 𝐄³ does not imply the orthogonality among 𝜕𝑥/𝜕𝑞, 𝜕𝑦/𝜕𝑞, and 𝜕𝑧/𝜕𝑞(=𝜕𝜑/𝜕𝑞). Therefore, 𝜕𝑧/𝜕𝑞 is not orthogonal to 𝐽^𝑇(𝑞)(=(𝜕𝑥/𝜕𝑞,𝜕𝑦/𝜕𝑞)), and hence the last term of the right hand side of (42) contains both components of image (𝑤) and ker(𝑤).

Now, substituting this into (27) yields𝐺(𝑞)̈𝑞+12𝐺+𝑆+𝐶̇̇𝑞+𝐽^𝑇=−Δ𝜆𝜕𝑧(𝑞)(𝑞)𝜁√𝑘𝐱+𝑘Δ𝐱̇𝜕𝑞.(43)Evidently the inner product of (43) and ̇𝑞 under the constraint 𝑧(𝑞)=𝜉 leads todd𝑡𝐸𝑞,̇𝑞=−̇𝑞𝐶̇𝑞−𝜁√𝑘‖‖𝐱̇‖‖²,(44)where𝐸𝑞,̇𝑞=12̇𝑞^𝑇𝐺(𝑞)̇𝑞+𝑘2‖Δ𝐱‖².(45)Unfortunately, this quantity is not positive definite in the tangent bundle 𝑇𝐹_𝜉. Nevertheless, it is possible to see that magnitudes of ̇𝑞(𝑡) and Δ𝐱(𝑡) remain small if at initial time 𝑡=0 both magnitudes ‖̇𝑞(0)‖ and ‖Δ𝐱(0)‖ are set small. Next, let us introduce the equilibrium manifold defined by the setEM₁=𝑞∶𝑧(𝑞)=𝜉,𝑥(𝑞)=𝑥_𝑑,𝑦(𝑞)=𝑦_𝑑,(46)which is of one dimension. Fortunately as discussed in the paper [9], it is possible to confirm that a modified scalar function𝑊_𝛼=𝐸𝑞,̇𝑞+𝛼̇𝑞^𝑇𝐺(𝑞)𝑃_𝜑𝐽^𝑇(𝑞)Δ𝐱(47)becomes positive definite in 𝑇𝐹_𝜉 with an appropriate positive parameter 𝛼>0, provided that the physical scale of the robot is ordinary, and feedback gains 𝐶 and 𝑘 are chosen adequately. Here, in (47), 𝑃_𝜑 is defined as 𝐼₄−𝑤𝑤^𝑇/‖𝑤‖². Furthermore, if in a Riemannian ball in 𝐹_𝜉 defined as 𝐵(𝑞(0),𝑟₀)={𝑞∶𝑑(𝑞,𝑞(0))<𝑟₀}, the Jacobian matrix 𝐽(𝑞) is nondegenerate, then it can be expected that there exist positive numbers 𝜎_𝑚 and 𝜎_𝑀 such that𝜎_𝑀𝐼₂≥𝐽𝐽^𝑇≥𝐽𝑃_𝜑𝐽^𝑇≥𝜎_𝑚𝐼₂.(48)Further, to simplify the argument, we choose 𝐶=𝑐𝐼₄ with a constant 𝑐>0. Then, it follows from differentiation of 𝑉(=̇𝑞^𝑇𝐺(𝑞)𝑃_𝜑𝐽^𝑇(𝑞)Δ𝐱) that𝑉=−̇−𝑐̇𝑞𝜁√_𝑇𝑘𝐱+𝑘Δ𝐱̇^𝑇𝐽𝑃_𝜑𝐽^𝑇Δ𝐱𝑃_𝜑𝐽^𝑇Δ𝐱+̇𝑞^𝑇𝐺𝑃_𝜑𝐽^𝑇𝐱−ℎ(̇𝑞,𝐺)Δ𝐱,̇(49)whereℎ(̇𝑞,𝐺)=̇𝑞^𝑇12𝐺𝑃̇^𝜑−𝑆𝑃_𝜑−𝐺𝑃_𝜑̇𝐽^𝑇−𝐺𝑃_𝜑𝐽̇^𝑇.(50)This 1×2-vector ℎ is quadratic in components of ̇𝑞, and each coefficient of _𝑖̇𝑞̇𝑞_𝑗 (for 𝑖,𝑗=1,…,4) is at most of order of the maximum spectre radius of 𝐺(𝑞) denoted by 𝑔_𝑀. Hence, it follows from (49) that||ℎ̇𝑞,𝐺Δ𝐱||≤𝑔_𝑀𝑙₀‖‖̇𝑞‖‖²‖Δ𝐱‖,(51)where 𝑙₀ signifies a constant that is of order of the maximum link length (because the Jacobian matrix 𝜕𝐱/𝜕𝑞^𝑇 is homogeneously related to the three link lengths of the robot shown in Figure 4). Further, note that ̇𝑞^𝑇𝑃_𝜑=0 in this case and remark thaṫ𝑞^𝑇𝐺𝑃_𝜑𝐽^𝑇𝐱≤̇12̇𝑞^𝑇𝐺𝐺̇𝑞+12𝐱̇^𝑇−𝜁√𝐽𝑃_𝜑𝐽^𝑇𝐱,̇(52)𝑘𝐱̇^𝑇𝐽𝑃_𝜑𝐽^𝑇Δ𝐱≤𝜁𝑘2Δ𝐱^𝑇𝐽𝑃_𝜑𝐽^𝑇Δ𝐱+𝜁2𝐱̇^𝑇𝐽𝐽^𝑇𝐱.̇(53)Thus, substituting (51) to (52) into (49) yields𝑉≤−̇𝑘−12−𝜁𝑘2Δ𝐱^𝑇+1𝐽𝑃_𝜑𝐽^𝑇Δ𝐱2̇𝑞^𝑇𝐺𝐺̇𝑞+1+𝜁2𝐱̇^𝑇𝐽𝐽^𝑇𝐱+𝑔̇_𝑀𝑙₀‖‖̇𝑞‖‖²‖Δ𝐱‖.(54)With the aid of a positive parameter 𝛼>0, it is easy to see that𝑊_𝛼≥𝐸−𝛼2̇𝑞^𝑇𝐺(𝑞)𝐺(𝑞)̇𝑞−𝛼2Δ𝐱^𝑇=1𝐽𝑃_𝜑𝐽^𝑇Δ𝐱2̇𝑞^𝑇{𝐺−𝛼𝐺𝐺}̇𝑞+12Δ𝐱^𝑇𝑘𝐼₂−𝛼𝐽𝑃_𝜑𝐽^𝑇Δ𝐱.(55)

Figure 4 

A hand-writing robot with four DOFs whose endpoint $P(=(x\mathrm{,}y))$ is constrained on a plane $z=\xi \mathrm{.}$

Now, suppose that the robot has an ordinary physical scale with𝑔_𝑀≤0.001[kgm²],𝑙₀≤0.2[m].(56)Then, it is possible to see that𝑊̇_𝛼=𝐸+𝛼̇𝑉≤−̇𝛼𝑘𝜎^𝑚2‖Δ𝐱‖²−𝑐3‖‖̇𝑞‖‖²,(57)provided that ‖Δ𝐱(𝑡)‖<0.3[m], (1+𝜁𝑘)≤𝑘, and 𝜁√𝑘>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 𝑘 is chosen between 5.0∼20.0[kg/s²] (see [9]). On account of (48) and (56),𝑊_𝛼≥1−𝛼𝛾₀𝐸,(58)and similarly,𝑊_𝛼≤1+𝛼𝛾₀𝐸,(59)where 𝛾₀=max(𝑔_𝑀,𝜎_𝑀/𝑘). Then, if the damping factor can be selected to satisfy both inequalities 𝑐≥(3/2)(𝛼𝜎_𝑚𝑔_𝑀) and 𝑐>𝛼𝑔_𝑀𝑙₀, then (57) is reduced to𝑊̇_𝛼≤−𝛼𝜎^𝑚1+𝛼𝛾₀𝑊_𝛼,(60)where 𝛾₀ can be set as 𝛾₀=𝜎_𝑀/𝑘 which is larger than 𝑔_𝑀. Hence, by choosing 𝛼=1/2𝛾₀=𝑘/2𝜎_𝑀, it follows from (60) and (59) that𝑊_𝛼(𝑡)=𝑊_𝛼(0)𝑒^−𝛾𝑡,(61)where 𝛾=𝑘𝜎_𝑚/3𝜎_𝑀.

Now, suppose that 𝑞^∗ in EM₁∩𝐵(𝑞(0),𝑟₀) is the minimizing point that connects with 𝑞(0) among all geodesics from 𝑞(0) to any point of EM₁∩𝐵(𝑞(0),𝑟₀). We call 𝑞^∗ a reference point corresponding to 𝑞(0).

Definition 1 (stable Riemannian ball on a submanifold). If for any 𝜀>0, there exists the number 𝛿(𝜀)>0 such that any solution trajectory (orbit) of (43) starting from an arbitrary initial position inside 𝐵(𝑞^∗,𝛿(𝜀)) with ̇𝑞(0)=0 remains inside 𝐵(𝑞^∗,𝜀) for any 𝑡>0, then the reference point 𝑞^∗ on EM₁ is said to be stable on a submanifold (see Figure 5).
It can be concluded from the exponential convergence of 𝑊_𝛼 to zero and the inequality 𝐸≤2𝑊_𝛼 when 𝛼=1/2𝛾₀ that any point inside 𝐵(𝑞^∗,𝛿(𝜀)) included in 𝐵(𝑞(0),𝑟₁) for some 𝑟₁(<𝑟₀) is stable on a submanifold and further such a solution trajectory converges asymptotically to some 𝑞^∞ 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 .

Figure 5 

Definition of the Riemannian ball $B(q^{*}\mathrm{,}\delta )$. Any orbit starting from $B(q^{*}\mathrm{,}\delta )$ converges to $\text{E}{\text{M}}_1\cap B(q^{*}\mathrm{,}\epsilon ).$

It should be noticed from the proof in Appendix that asymptotic convergence of the solution trajectory to some 𝑞^∞ on the equilibrium manifold implies also the asymptotic convergence of constraint force 𝜆(𝑡) to 𝜆_𝑑 as 𝑡→∞ because Δ𝜆=𝜆−𝜆_𝑑 is expressed asΔ𝜆(𝑡)=1𝑤^𝑇𝐺⁻¹𝑤_𝑘𝑤̇𝑞̇^𝑘−_𝑖,𝑗𝑤^𝑘Γ^𝑘𝑖𝑗̇𝑞^𝑖̇𝑞^𝑗−𝑤_𝑇𝐽^𝑇𝜁√𝑘𝐱+𝑘Δ𝐱̇,(62)and this right hand side converges to zero as 𝑡→∞.

The stability notion of a Riemannian ball in a neighborhood of a reference equilibrium state 𝑞^∗ on EM₁ is extended to cope with the case that the initial velocity vector ̇𝑞(0) is not zero. To do this, we define an extended Riemannian ball in the tangent bundle 𝑀×𝑇𝐹_𝜉 around (𝑞^∗,̇𝑞=0) in such a way that𝐵=𝑞^∗,0𝑞,̇𝑞;𝑟₀∶𝑑,𝑟_𝐾𝑞,𝑞_∗<𝑟₀,(1/2)̇𝑞^𝑇𝐺(𝑞)̇𝑞<𝑟_𝐾,(63)where 𝑑(𝑞,𝑞^∗) denotes the distance between 𝑞 and 𝑞^∗ restricted to the submanifold 𝐹_𝜉, and 𝐺 is defined below (40).

Definition 2 (asymptotic stability on a submanifold). If for any 𝜀>0, there exist numbers 𝛿(𝜀) and 𝛿_𝐾(𝜀) such that any solution trajectory of (43) starting from an arbitrary initial position and velocity inside 𝐵{(𝑞^∗,0);(𝛿(𝜀),𝛿_𝐾(𝜀))} remains in 𝐵{(𝑞^∗,0);(𝜀,𝑟_𝐾)} and further converges asymptotically to some equilibrium point 𝑞^∞∈EM₁ 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 𝑥𝑦-plane 𝐄². We denote the object mass center by 𝑂_𝑚 with the coordinates (𝑥_𝑚,𝑦_𝑚) expressed in the inertial frame. On the other hand, we express a local coordinate system fixed at the object by 𝑂_𝑚-𝑋𝑌 together with unit vectors 𝐫_𝑋 and 𝐫_𝑌 along the 𝑋-axis and 𝑌-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),𝑌(𝑠)) in terms of arc length parameter 𝑠 as shown in Figure 7.

Figure 6 

Two-dimensional object grasping by a pair of multijoint robot fingers.

Figure 7 

Definitions of physical symbols in relation to rolling contact. ${\theta }_1$ denotes ${\theta }_1(s)$ appearing in (64).

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),𝑌(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:𝜃₁(𝑠)=arct𝑋^(𝑠)𝑌^(𝑠),(64)where 𝑋^(𝑠)=d𝑋(𝑠)/d𝑠 and 𝑌′=d𝑌(𝑠)/d𝑠. In this paper, all angles are set positive in counterclockwise direction. Then,𝑃₁𝑃^1=𝑙_𝑛1(𝑠)=−𝑋(𝑠)cos𝜃₁(𝑠)+𝑌(𝑠)sin𝜃₁(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):𝑃₁𝑃^1=−𝑥_𝑚𝑦−𝑥_01𝑚cos𝜃+𝜃₁(𝑠)−𝑦₀₁sin𝜃+𝜃₁(𝑠)−𝑟₁.(66)Hence, the left-hand contact constraint can be expressed by the holonomic constraint𝑄₁=−+𝑥𝑦_𝑚𝑚−𝑥₀₁cos𝜃+𝜃₁=0,−𝑦₀₁sin𝜃+𝜃₁+𝑟₁+𝑙_𝑛1(𝑠)(67)where 𝑙_𝑛1(𝑠) denotes the right-hand side of (65) and 𝜃₁=𝜃₁(𝑠) for abbreviation.

Figure 8 

Physical and geometrical meanings of key variables $l_{n1}(s)$ and $l_{e1}(s)$ defined in relation to the object contour $c(s)$ expressed in terms of length parameter $s$ of the curve.

Next, we note that the length 𝑂_𝑚𝑃^1 can be regarded also as a shape function of the object given by𝑂_𝑚𝑃^1=𝑙_𝑒1(𝑠)=𝑋(𝑠)sin𝜃₁+𝑌(𝑠)cos𝜃₁.(68)On the other hand, this quantity can be also expressed as𝑂_𝑚𝑃^1=−=𝑌₁(𝑡)𝑥_𝑚−𝑥₀₁sin𝜃+𝜃₁−𝑦_𝑚−𝑦₀₁cos𝜃+𝜃₁.(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=𝑞₁₁+𝑞₁₂+𝑞₁₃, we have−𝑟=−₁̇𝑝₁𝑌+𝑟₁̇₁𝜃̇=+̇𝑥_𝑚−̇𝑥𝑥_01𝑚sin𝜃+𝜃₁+̇𝑦_𝑚−̇𝑦₀₁cos𝜃+𝜃₁−𝑥₀₁cos𝜃+𝜃₁−𝑦_𝑚−𝑦₀₁sin𝜃+𝜃₁)𝜃,̇(70)which from (67) can be reduced to−𝑟₁+̇𝑝₁−𝑙̇𝑦_𝑛101𝜃+̇̇𝑥₀₁−̇𝑥sin𝜃+𝜃₁−̇𝑦cos𝜃+𝜃₁=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 𝑄_𝑖=0(𝑖=1,2), it is possible to construct a Lagrangian:𝐿=12̇𝑞^𝑇𝑖𝐺_𝑖𝑞_𝑖̇𝑞_𝑖+12𝑀̇𝑥²−𝑓₁+𝑀̇𝑦²+𝐼𝜃̇²𝑄₁−𝑓₂𝑄₂,(72)where 𝑞_𝑖 denote the joint vector for finger 𝑖, 𝐺_𝑖(𝑞_𝑖) 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 𝜆_𝑖(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:𝑀𝐼𝜃−𝑓̈̈𝑥̈𝑦−𝑓₁𝑌₁₁+𝑓₂𝑌₂−𝜆₁𝑙_𝑛1+𝜆₂𝑙_𝑛2=0,(73)𝐧₁−𝑓₂𝐧₂−𝜆₁𝐞₁−𝜆₂𝐞₂=0,(74)𝐺_𝑖𝑞_𝑖̈𝑞_𝑖+12𝐺̇^𝑖+𝑆_𝑖̇𝑞^𝑖+𝑓_𝑖𝐽^𝑇𝑖𝑞_𝑖𝐧_𝑖+𝜆_𝑖𝐽^𝑇𝑖𝑞_𝑖𝐞_𝑖+(−1)^𝑖𝑟_𝑖𝐩_𝑖=𝑢_𝑖,𝑖=1,2,(75) where 𝐩₁=(1,1)^𝑇, 𝐩₂=(1,1,1)^𝑇, and 𝐽^𝑇𝑖(𝑞_𝑖)=𝜕(𝑥_0𝑖/𝑦_0𝑖)/𝜕𝑞_𝑖, and𝐧_𝑖=(−1)^𝑖−cos𝜃+𝜃^𝑖sin𝜃+𝜃_𝑖,𝐞^𝑖=sin𝜃+𝜃^𝑖cos𝜃+𝜃_𝑖(76)for 𝑖=1,2. It is important to note that the object dynamics of (73) and (74) can be recast in the form𝐻₀𝐳+𝑓̈₁𝐰₁+𝑓₂𝐰₂+𝜆₁𝐰₃+𝜆₂𝐰₄=0,(77)where 𝐳=(𝑥,𝑦,𝜃), and𝐰_𝑖=−𝐧^𝑖(−1)^𝑖𝑌_𝑖,𝐰^𝑖+2=−𝐞^𝑖(−1)^𝑖𝑙_𝑛𝑖,𝑖=1,2,(78)which are of a two-dimensional wrench vector. This implies that if the sum of all wrench vectors converges to zero, then the force/torque balance is established. Further, if we define𝐗=𝐮=0,0,0,𝑢𝑥,𝑦,𝜃,𝑞^𝑇1_𝑇1,𝑞^𝑇2^𝑇,𝐺(𝐗)=diag𝑀,𝑀,𝐼,𝐺,𝑢^𝑇2₁^𝑇,𝑆(𝐗,𝐗)=diaġ0,0,0,𝑆𝑞₁₁,𝐺₂𝑞₂,,𝑆₂,(79)then (73) ∼ (75) can be written in the form𝐺(𝐗)𝐗+̈12𝐺+𝑆̇𝐗+̇_𝑖=1,2𝑓^𝑖𝜕𝑄_𝑖𝜕𝐗+𝜆^𝑖𝜕𝑅_𝑖𝜕𝐗=𝐮,(80)where𝑅_𝑖=−(−1)^𝑖𝑟_𝑖𝜃+𝜃_𝑖𝑠_𝑖−𝐩^𝑇𝑖𝑞_𝑖+𝑌_𝑖+𝜓_𝑖𝑠_𝑖,(81)and 𝜓_𝑖 signifies a function depending only on the shape parameter 𝑠_𝑖. The details of the integral form of (81) will be discussed in Appendix . Comparing (80) with (27), it must be understood that a similar but extended argument of Section 4 can be applied even for a case of physical interaction with complex holonomic constraints and nonholonomic (but Pfaffian) constraints. More precisely, when 𝐮=0 in (80), the image space of the Riemannian manifold (𝐗,𝐺(𝐗)) with constraints 𝑄₁=0 and 𝑄₂=0 is spanned from the gradients𝐺⁻¹(𝐗)𝜕𝑄¹𝜕𝐗,𝜕𝑄𝜕𝐗²,𝜕𝑅𝜕𝐗¹,𝜕𝑅𝜕𝐗²,(82)and the kernel space should be defined as the orthogonal compliment to the image space. Note that the rolling constraints of (71) and another for 𝑖=2 induce further restriction of the tangent space to a more subdimensional linear space. Here, 𝜕𝑅_𝑖/𝜕𝐗 should be taken at the time instant