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

A Riemannian-geometry approach for modeling and control of dynamics of object manipulation under holonomic or nonholonomic 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.


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-or 3-) 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 firstorder 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.

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 formula where q = (q 1 , q 2 ) T denotes the vector of joint angles, H(q) denotes the inertia matrix, S(q, q) q 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 H 0 such that for any q.It should be also noted that S(q, q) is skew symmetric and linear and homogeneous in q.More in detail, the i jth entry of S(q, q) denoted by s i j can be described in the form [3] where H(q) = (h i j (q)), from which it follows apparently that s i j = −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).When two joint angles θ 1 and θ 2 are given in θ i ∈ (−π, π], i = 1, 2, for the 2 DOF robot arm shown in Figure 1, the posture p(θ 1 , θ 2 ) 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 = (θ 1 , θ 2 ) and any number α > 0, a set of all p = (θ 1 , θ 2 ) is defined as where 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 R 2 .Such a homeomorphism φ : In fact, a neighborhood U p,α of posture p with joint angles (θ 1 , θ 2 ) in Figure 1 can be mapped to an open set Ω in R 2 with 2 dimensional numerical coordinates (q 1 , q 2 ) 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 R 3 (see Figure 3) in which angles q 1 and q 2 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 (q 1 , . . ., q n ) can be identically chosen as a set of n independent joint angles (θ 1 , . . ., 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 ∞ .Now, it is necessary to define a tangent vector to an abstract differentiable manifold M at p ∈ M. Let I be an interval (− , ) 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 where symbol () means differentiation of () with respect to t ∈ I.This definition of tangent vectors to M at p does not q2 y x z O q 1 (q 2 = 0) 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 p M 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 : 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 be 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 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( 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 p there exists at least a minimizing curve c(t 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)) = (q 1 (t), . . ., q n (t)), then q(t) = φ(c(t)) satisfies the 2nd-order differential equation where Γ k i j denotes Christoffel's symbol defined by and (g kh ) denotes the inverse of matrix (g kh ).A curve q(t) : is called the energy of the curve.Then, by applying the Cauchy-Schwartz inequality for (7), we have Further, the equality of (11) 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.

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 form 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 = (θ 1 , . . ., 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 = (q 1 , . . ., 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 S 1 : 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 form Hence, the equation of motion of the robot is expressed by Lagrange's equation d dt where L(q, q) = K(q, q), and u stands for a generalized external force vector.It is interesting to note that in differential geometry, (15) can be described as where Γ ik j denotes Christoffel's symbol of the first kind defined by For later use, we introduce another Christoffel's symbol called the second kind as shown in the formula 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 ik j = Γ jki and Γ k i j = Γ k ji .Now, we show that ( 13) is equivalent to ( 16) by bearing in mind that Ḣ(q) = i {∂H(q)/∂q i } qi , and the skew symmetric matrix S(q, q) 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 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 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 i j (q) instead of h i j (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 where v = v i (∂/∂q i ) ∈ T p M and w = w j (∂/∂q j ) ∈ T p M, and the summation symbol in i and j is omitted, and q = (q 1 , . . ., 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, where 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 have where the equivalent expression G(q) q + 1 2 Ġ(q) + S q, q q = 0 (24) to ( 13) with u = 0 is used, G(q) = (g i j (q)), and s i j of S is given in the form of (3) (where h i j = g i j ).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 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 = (q 1 , . . ., q n ) T .At the same time, it is important to note that, in differential geometry, choice of coordinates in the tangent space T p M 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 = (q 1 , . . ., q n ) T , then the coordinates in the tangent space T p M should be chosen as the vector of joint angular velocities ∂/∂q = (∂/∂q 1 , . . ., ∂/∂q n ) correspondingly to q = ( q1 , . . ., qn ) T , from which the Riemannian metric g i j (q) is defined through the inertia matrix.

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 as 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 and consider the minimization 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 equation together with the constraint condition ϕ(x(t)) = ξ, where 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 Since the holonomic constraint ϕ(x(q)) = ξ implies w, q = 0, it follows that Substituting this into (32), we obtain 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 torques Then, by substituting this into (27), we obtain the Lagrange equation of motion under the constraint ϕ(x(q)) = ξ: where 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 d J 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 transformation where P is a 4 × 3 matrix whose column vectors with the unit norm are orthogonal to w, and η denotes a 3 × 1 matrix (3dim.vector) and ż a scalar.Since Q is an orthogonal matrix, Restriction of (36) to the kernel space of w can be attained by multiplying (36) by P T from the left such that which is reduced to the Eular equation in η: or equivalently where G(q) = P T G(q)P, Γ k i j the Christoffel symbol for (g i j ) = G, and which is skew symmetric, too.Note that the transformation Q is isometric, and (40) stands for the geodesic equation on the constraint Riemannian submanifold.

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 form where ).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 E 3 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) yields Evidently the inner product of (43) and q under the constraint z(q) = ξ leads to where 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 set which is of one dimension.Fortunately as discussed in the paper [9], it is possible to confirm that a modified scalar function 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 I 4 − ww T / w 2 .Furthermore, if in a Riemannian ball in F ξ defined as B(q(0), r 0 ) = {q : d(q, q(0)) < r 0 }, the Jacobian matrix J(q) is nondegenerate, then it can be expected that there exist positive numbers σ m and σ M such that Further, to simplify the argument, we choose C = cI 4 with a constant c > 0.Then, it follows from differentiation of V (= qT G(q)P ϕ J T (q)Δx) that where This 1 × 2-vector h is quadratic in components of q, and each coefficient of qi qj (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 where l 0 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 qT P ϕ = 0 in this case and remark that qT Thus, substituting (51) to (52) into (49) yields With the aid of a positive parameter α > 0, it is easy to see that Now, suppose that the robot has an ordinary physical scale with Figure 5: Definition of the Riemannian ball B(q * , δ).Any orbit starting from B(q * , δ) converges to EM 1 ∩ B(q * , ε).
Then, it is possible to see that 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/s 2 ] (see [9]).On account of (48) and (56), and similarly, where γ 0 = max(g M , σ M /k).Then, if the damping factor can be selected to satisfy both inequalities c ≥ (3/2)(ασ m g M ) and c > αg M l 0 , then (57) is reduced to where γ 0 can be set as γ 0 = σ M /k which is larger than g M .Hence, by choosing α = 1/2γ 0 = k/2σ M , it follows from (60) and (59) that where γ = kσ m /3σ M .Now, suppose that q * in EM 1 ∩B(q(0), r 0 ) is the minimizing point that connects with q(0) among all geodesics from q(0) to any point of EM 1 ∩ B(q(0), r 0 ).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 EM 1 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 = 1/2γ 0 that any point inside B(q * , δ(ε)) included in B(q(0), r 1 ) for some r 1 (< r 0 ) is stable on a submanifold and further such a solution trajectory converges asymptotically to some q ∞ on the equilibrium manifold EM 1 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 A.
It should be noticed from the proof in Appendix A 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 as 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 EM 1 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 that B q * , 0 ; r 0 , r K = q, q : d q, q * < r 0 , (1/2) qT G(q) q < r K , 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 ∞ ∈ EM 1 with still state, then the reference point with its posture on EM 1 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 inertiaoriginated 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].

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 E 2 .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.
First, suppose that at the left-hand contact point P 1 the fingertip of the left finger is contacting with the object.Denote the unit normal at P 1 by n 1 and the unit tangent vector by e 1 .Note that n 1 is normal to both the object surface and finger-end sphere at P 1 , and e 1 is tangent to them at P 1 , too.If we denote position P 1 by local coordinates (X(s), Y (s)) fixed at the object (see Figure 7), then the angle from the Xaxis to the unit normal n 1 is assumed to be determined by a function on the curve: where X (s) = dX(s)/ds and Y = dY (s)/ds.In this paper, all angles are set positive in counterclockwise direction.Then, 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): Hence, the left-hand contact constraint can be expressed by the holonomic constraint where l n1 (s) denotes the right-hand side of (65) and θ 1 = θ 1 (s) for abbreviation.
Next, we note that the length O m P 1 can be regarded also as a shape function of the object given by On the other hand, this quantity can be also expressed as As discussed in [13], in the light of the book [14], pure rolling at P 1 is defined by the condition that the translational velocity of P 1 on the finger sphere is equal to that of P 1 on the object contour along the e 1 -axis.Hence, by denoting p 1 = q 11 + q 12 + q 13 , we have which from (67) can be reduced to This constraint form is Pfaffian.As to the contact point P 2 at the right-hand finger end, a similar nonholonomic constraint can be obtained.Thus, by introducing Lagrange's multipliers f 1 and f 2 associated with holonomic constraints , it is possible to construct a Lagrangian: 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 P 2 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: where p 1 = (1, 1) T , p 2 = (1, 1, 1) T , and J T i (q i ) = ∂(x 0i /y 0i )/∂q i , and for i = 1, 2. It is important to note that the object dynamics of ( 73) and ( 74) can be recast in the form where z = (x, y, θ), and 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 define then (73) ∼ (75) can be written in the form where 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 B. 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 Q 1 = 0 and Q 2 = 0 is spanned from the gradients 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 t under the condition that s i are fixed.Nevertheless, as shown in Appendix B, it is quite interesting to know that R i for i = 1, 2 can be regarded as a constant function in change of t even under any infinitesimally small variation of s i , that is, dR i /dt = 0 for i = 1, 2. In other words, R i = constant (i = 1, 2), and these expressions imply a holonomic constraint.Such a geometric structure of decomposition of the tangent space to an image space of constraint gradients and its orthogonal compliment is known in general mathematical terminology (see [15,16]) but has not yet been critically spelled out in the case of existence of rolling contact connections between curved rigid bodies.The Euler equation of (80) can be expressed in a similar form to (1) by introducing the constraint vector Φ and the vector of Lagrange multipliers λ as Then, (80) can be written in the form which must be valid for the constraint ẊT (∂Φ/∂X) = 0. We denote the (n − 4)-dimensional kernel space of ∂Φ/∂X by V X and its 4-dimensional orthogonal compliment as the image space of G −1 (X)∂Φ/∂X by H X .
We are now in a position to derive an update law of the length parameters s 1 and s 2 along the object contours.Firstly, it is important to verify that the holonomic constraint Q 1 = 0 is invariant under any infinitesimally small variation of s 1 .In fact, we see from (64) to (69) that where we denote s 1 = s for simplicity.Next, it is possible to show that if θ 1 (s) is variable, then θ 1 (s) = dθ 1 (s)/ds = κ 1 (s), where κ 1 (s) is the curvature of the left-hand contour of the object.Then, it is obvious from Figures 6 to 8 that where κ 1 (s) / = 0. Since θ 1 (s) = κ 1 (s), ( 86) is reduced to This equation should be accompanied with the Euler-Lagrange equation ( 80).This can be also regarded as a nonholonomic constraint to (80).As to the right-hand contour, it follows that ds 2 dt

Lifting in Horizontal Space and Force/Torque Balance
First, in order to find an adequate lifting that belongs to the image space H X and realizes the force/torque balance (see (77)) in the sense that we remark that where We also define Then, it follows from (90) and (91) that where s = (s 1 , s 2 ) and (95) Thus, let us define and remark that they satisfy Further, note that where This shows according to (98) that if S N tends to vanish, then the force/torque balance is established, that is, the total sum of wrench vectors exerted to the object becomes zero.

Control Signal for Blind Grasping and a Morse-Lyapunov Function
From the practical standpoint of designing a control signal for stable grasping, it is important to see that objects to be grasped are changeable, but the pair of robot fingers is always the same.That is, for designing control signals, we are unable to use physical parameters of the object such as the location (x m , y m ) of its mass center and object geometry.On the contrary, it is possible to assume the knowledge of finger kinematics like finger link lengths and locations of the centers of finger-end spheres and to use measurement data of finger joint angles and angular velocities.In view of these points, let us propose a family of control signals defined as where and p 1 = (1, 1, 1) T , p 2 = (1, 1) T , c i denotes a damping factor, and γ i > 0 a positive gain specified later.The closed loop dynamics of motion of the overall fingers-object system can be derived by substituting u i of (100) into (75) for i = 1, 2. In order to spell out the dynamics in a more physically meaningful way for later discussions, first note the following three equalities: Then, by substituting (98), ( 102) into (73) to (75), it is possible to express the closed loop dynamics in the following forms: Similarly, to the form of (84), these equations can be written in the following way: where b θ = (0, 0, 1, 0, 0, 0, 0, 0) T , b 1 = (0, 0, 0, 1, 1, 1, 0, 0) T , b 2 = (0, 0, 0, 0, 0, 0, 1, 1) T . ( At this stage, it is important to note that in accordance with four constraints Φ = 0, the velocity vector Ẋ belongs to the kernel space of (∂Φ/∂X) T and, therefore, ẊT ∂Φ/∂X = 0. Further by using (100) and taking inner product of (84) and Ẋ, we obtain where K denotes the system's kinetic energy defined by The relation of ( 106) must be equivalently derived by taking inner product of Ẋ and the closed loop dynamics of equation (104).To verify this, let us define p 1 = q 11 + q 12 + q 13 = p T 1 q 1 , p 2 = q 21 + q 22 = p T 2 q 2 , (108) where l and s are defined in (95).It should be remarked that, with the aid of expressions of integral form of rolling constraints shown in (71), we have and hence, Similarly, from (66), (69), and (101), we have Thus, the inner product of Ẋ and (104) or the relation of ( 106) is reduced to In other words, the closed loop dynamics of (104) can be expressed in the form This is interpreted as a Lagrange equation of the Lagrangian in accompany with the external damping torques c i qi for i = 1, 2 through finger joints.The scalar function P defined by ( 109) is a quadratic function of Y 1 , Y 2 , p 1 , and p 2 , and hence, it is regarded as a quadratic function of θ, p 1 , and p 2 since Y i can be regarded as a linear function of θ and p i for i = 1, 2 because of (71).Hence, P can be regarded as a Morse function defined on the Riemannian submanifold induced by two constraints described by Q i = 0 for i = 1, 2 and four constraints in the tangent space described by Qi and Ṙi = 0 for i = 1, 2.

Physical Insights into Gradient and Hessian of the Morse Function
The physical meaning of control signals for blind grasping defined by ( 100) is quite simple.The first term of the righthand side of (100) plays a role of damping for rotational motion of finger joints.Damping for motion of the object is exerted from velocity constraints Qi = 0 and Ṙi = 0 for i = 1, 2 as discussed in detail in the previous paper [1].The second term plays a role of fingers-thumb opposition that induces minimization of the distance between O 01 and O 02 , centers of finger-end spheres.The distance is equivalent to √ l 2 + d 2 as discussed in Section 5.The third term plays an important role in suppressing excess movements of rotation of finger joints.These characteristics of the control signal condense into the Lagrange equation of motion with (1) the potential P(s, θ, p 1 , p 2 ) of (109), (2) the lifting (∂Φ/∂X)λ d , where λ d = ( f 1d , f 2d , λ 1d , λ 2d ) T , and (3) the gradient ∂P/∂X of the potential.In other words, (113) implies that if the artificial potential P attains its minimum and, at the same time, makes the gradient ∂P/∂X vanish at some s = s * (s = (s 1 , s 2 ) T ), and, moreover, the artificial potential is positive definite in X under the two holonomic constraints, then the equilibrium position that minimizes P would be asymptotically stable.Unfortunately, P is a quadratic function with respect to only θ, p 1 , and p 2 , and, therefore, P is only nonnegative definite in X.Nevertheless, it is possible to show that the Hessian matrix of P with respect to θ, p 1 , and p 2 becomes positive definite in these three variables as shown in Table 1 that is calculated by partially differentiating the gradients ∂P/∂θ, ∂P/∂p 1 , and ∂P/∂p 2 of (111) to (112) with respect to θ, p 1 , and p 2 again.Apparently, the Hessian becomes positive definite provided that γ i is chosen as being of similar order of (r i / f d ), and θ i (s i ) remains in a region such that |θ i (s i )| < π/6 for i = 1, 2. Then, by using a similar argument used in proving the stability on a submanifold for DOF redundant systems [9,13], a solution trajectory to the Lagrange equation converges exponentially to the constraint equilibrium manifold.In this paper, we omit the details of the argument, but show a physical meaning of the gradient ∂P/θ together with ∂P/∂p i (i = 1, 2).
As shown in Figure 9, all rotational motions of the object emerge around axes that are perpendicular to the xy-plane.Here, we consider the rotational moment that may emerge at the contact point P 2 exerted by the pressing force f 1 to the object from the other contact point P 1 .Another force f 2 at P 2 that presses the object generates the torque around the zaxis at P .The sum of these torques around the z-axis can be expressed as where we denote X i = X(s i ) and Y i = Y (s i ) for abbreviation.We see also that from geometric relations of the vectors − −− → O m P i , and quantities Y i and l ni expressed in local coordinates O m -XY (see Figure 9), it follows that (118) As a summary of the argument, it is possible to conclude that a solution trajectory to (114) converges asymptotically to some equilibrium state X = X * with some s = s * with ∂P/∂X = 0 at X = X * and s = s * .

Conclusions
A natural extension of hybrid position/force control for robots with redundant degrees of freedom is presented from the standpoint of a Riemannian geometric approach.It is shown that any supply of the constant pressing force lying in the image space of the constraint gradient to the environment through joint actuations is not relevant to motions in the kernel space orthogonally compliment to the image space.An extension of problems of grasping and manipulation of rigid objects to the case of 2-dimensional objects with arbitrary shape is also treated from the viewpoint of Dirichlet-Lagrange's stability by introducing a nonnegative definite Morse-Lyapunov function on a Riemannian manifold together with damping shaping.

Figure 4 :
Figure 4: A hand-writing robot with four DOFs whose endpoint P(= (x, y)) is constrained on a plane z = 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 as

Figure 6 :
Figure 6: Two-dimensional object grasping by a pair of multijoint robot fingers. y

Figure 8 :
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.

Figure 9 :
Figure9: The pressing force to the object at P 1 in the direction n 1 induces a rotational moment around P 2 and vice versa.

Table 1 :
Hessian matrix of the potential P.
1) i l ei s i + s i , i = 1, 2. (B.2)In fact, let us define in accordance with (81) and (B.2)R i = r i θ + θ i s i − p i − (−1) i Y i − l ei s i + s i (B.3)for i = 1, 2 and take the derivative of them in t.Then, it follows that dR i dt = r i θ − ṗi − (−1) i ∂Y i ∂q T i X s i sin θ i + Y s i ) cos θ i + X s i cos θ i − Y s i sin θ = (−1) i x m − x 0i cos θ + θ i + y m − y 0i sin θ + θ