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 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.
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 -dimensional configuration space. Planning of robot motions has been investigated traditionally on the basis of kinematics on a configuration space as an -dim numerical space .
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 (, ) associated with the Riemannian metric 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 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 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 formulawhere denotes the vector of joint angles, denotesthe inertia matrix, the gyroscopic force term including centrifugal and Coriolis forces, the gradient vector of a potential function due to the gravity with respect to , and the joint torque generated by joint actuators . It is well known that the inertia matrix is symmetric and positive definite, and there exist a positive constant together with a positive definite constant diagonal matrix such thatfor any . It should be also noted that is skew symmetric and linear and homogeneous in . More in detail, the th entry of denoted by can be described in the form where , from which it follows apparently that . 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 is a constant or a sinusoidal function of components of joint angle vector . That is, each element of and is differentiable of class (infinitely differentiable in ).
When two joint angles and are given in , , for the 2 DOF robot arm shown in Figure 1, the posture is determined naturally. Denote the set of all such possible postures by and introduce a family of subsets of such that, for any with joint angles and any number , a set of all is defined aswherecan be regarded as an open subset of . Then, the set with such a family of open subsets can be regarded as a topological manifold. It is possible to show that the manifold becomes Hausdorff and compact. Further, every point of has a neighborhood that is homeomorphic to an open subset of 2-dimensional numerical space . Such a homeomorphism is called a coordinate chart. In fact, a neighborhood of posture with joint angles in Figure 1 can be mapped to an open set in with 2 dimensional numerical coordinates with the origin (Figure 2). In this case, it is possible to see that the original set of robot postures can be visualized as a torus shown in (see Figure 3) in which angles and 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 can be identically chosen as a set of independent joint angles by setting . 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 . As discussed in detail in mathematical text books [5, 6], the topological manifold of such a torus can be regarded as a differentiable manifold of class .
Now, it is necessary to define a tangent vector to an abstract differentiable manifold at . Let be an interval and define a curve by a mapping such that . A tangent vector to at is an equivalence class of curves for the equivalence relation defined by
if and only if, in a coordinate chart around , we have , where symbol means differentiation of with respect to . This definition of tangent vectors to at does not depend on choice of the coordinate chart at , as discussed in text books [5, 6]. Let us denote the set of all tangent vectors to at by and call it the tangent space at . It has an -dimensional linear space structure like . We also denote the disjoint union of the tangent spaces to at all the points of by and call it the tangent bundle of .
Now, we are in a position to define a Riemannian metric on a differentiable manifold as a mapping such that is of class and for and is a symmetric positive definite quadratic form
Suppose that is a connected Riemannian manifold. If is a curve segment of class , we define the length of to bewhere we assume for any and call such a curve segment to be regular. A mapping of class is called a piecewise regular curve segment (for brevity, we call it an admissible curve) if there exists a finite subdivision such that for is a regular curve for . Then, it is possible to define for any pair of points the Riemannian distance to be the infimum of all admissible curves from to . It is well known [4, 5] that, with the distance function defined above, any connected Riemannian manifold becomes a metric space whose induced topology is coincident with the given manifold topology. An admissible curve in a Riemannian manifold is said to be minimizing if for any other admissible curve with the same endpoints. It follows immediately from the definition of distance that is minimizing if and only if is equal to the distance between its endpoints. Further, it is known that if the Riemannian manifold is complete, then for any pair of points and there exists at least a minimizing curve , , with and . If such a minimizing curve is described with the aid of coordinate chart as then satisfies the 2nd-order differential equationwhere denotes Christoffel's symbol defined byand denotes the inverse of matrix . A curve satisfying (8) together with is called a geodesic, and (8) itself is called the Euler-Lagrange equation or the geodesic equation.
Given a -class curve , the quantityis called the energy of the curve. Then, by applying the Cauchy-Schwartz inequality for (7), we haveFurther, the equality of (11) follows if and only if is constant. It is also possible to see that if is a geodesic with and , then for any other curve with the same endpoints, it holdsThe equalities hold if and only if is also a geodesic. Conversely, if with and is a curve that minimizes the energy and makes constant, then becomes a geodesic connecting and . In mechanics, is usually called “action of ,” and is considered as the orbit of motion of a multibody system.
3. Riemannian Geometry of Robot Dynamics
Dynamics of a robotic mechanism with 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 -axis that is perpendicular to the -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 formwhere 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 (included in (1)) can be compensated by joint actuators through control input . In general, we can represent a posture of the robot as a physical entity by a family of joint angles , which can be expressed by a point in the -dimensional numerical space . In fact, we can naturally imagine an isomorphism , where and is an open subset of . In other words, a local coordinate chart can be treated to be identical to itself, an open subset of , by regarding (“” denotes transpose and hence a column vector) identical to by setting . In this way, the abstract manifold as the set of all robot postures can be regarded as an -dimensional torus as an -tuple direct product of . Hence, a robot posture can be represented by a point on and also expressed by a joint vector in .
From the definition of inertia matrices, is symmetric and positive definite, and the kinetic energy is expressed as a quadratic formHence, the equation of motion of the robot is expressed by Lagrange's equationwhere and stands for a generalized external force vector. It is interesting to note that in differential geometry, (15) can be described aswhere denotes Christoffel's symbol of the first kind defined byFor later use, we introduce another Christoffel's symbol called the second kind as shown in the formulawhere denotes the inverse of , the inertia matrix . Since and are symmetric, it follows that and . Now, we show that (13) is equivalent to (16) by bearing in mind that , and the skew symmetric matrix 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) thatSubstituting 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 yieldsIf , this expression is nothing, but the Euler-Lagrange equation shown in (8). By this reason, from now on, we use symbol instead of for the inertia matrix even when robot dynamics are treated.
Now, on the abstract topological manifold 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 and , and the summation symbol in and is omitted, and represents local coordinates. Then, the manifold can be regarded as a Riemannian manifold and becomes complete as a metric space. Then, according to the Hopf-Rinow theorem , any two points can be joined by a geodesic of length , that is, a curve satisfying (8) with shortest length, wherewith and .
As discussed in the previous section, geodesics are the critical points of the energy functional . Further, a geodesic curve satisfies In fact, by regarding that is an orbit on , we havewhere the equivalent expressionto (13) with is used, , and of is given in the form of (3) (where ). It is also important to note that, on a local coordinate chart corresponding to a neighborhood of , an orbit parameterized by time and expressed by a solution to (20) (where is of in ) should satisfyoras long as , where . When , and then the curve connecting and 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 . 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 -dimensional torus is one of such choice of local coordinates corresponding to the choice of joint angles . At the same time, it is important to note that, in differential geometry, choice of coordinates in the tangent space is indeterminable or free to choose. However, once a local coordinates system for an degrees of freedom robot is chosen by joint angle vector , then the coordinates in the tangent space should be chosen as the vector of joint angular velocities correspondingly to , from which the Riemannian metric is defined through the inertia matrix.
4. Constraint Submanifold and Hybrid Position/Force Control
Consider an -DOF robotic arm whose last link is a pencil and suppose that the endpoint of the pencil is in contact with a flat surface , where . It is well known that the Lagrange equation of motion of the system is written aswhere can be decomposed into and . On the constraint manifold , let us consider a smooth curve that connects the given two points and where and belong to . The length of such a curve constrained to is defined asand consider the minimizationthat should be called the distance between and on the constraint manifold. Then, the minimizing curve called the geodesic denoted identically by must satisfy the Euler equationtogether with the constraint condition , whereand . It should be noted that, from the inner product of (30) and it follows thatSince the holonomic constraint implies , it follows thatSubstituting this into (32), we obtainFrom the Riemannian geometry, the constraint force with the stands for a component of the image space of that is orthogonal to the kernel of . 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, should be regarded as a magnitude of the constraint force that presses the surface 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 torquesThen, by substituting this into (27), we obtain the Lagrange equation of motion under the constraint :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 positive. In a mathematical sense, exertion of the joint torque 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 . 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 transformationwhere is a matrix whose column vectors with the unit norm are orthogonal to and denotes a matrix (3-dim. vector) and a scalar. Since is an orthogonal matrix, . Hence, if then . Restriction of (36) to the kernel space of can be attained by multiplying (36) by from the left such thatwhich is reduced to the Eular equation in :or equivalentlywhere , the Christoffel symbol for , andwhich is skew symmetric, too. Note that the transformation 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 formwhere , , . 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 as matrix because we consider a special case of , that is, the constraint , and solution trajectories that satisfy . In relation to this, denote also the two-dimensional vector by and by . It should be noted that the orthogonality relationship among coordinates in does not imply the orthogonality among , , and . Therefore, is not orthogonal to and hence the last term of the right hand side of (42) contains both components of image and .
Now, substituting this into (27) yieldsEvidently the inner product of (43) and under the constraint leads towhereUnfortunately, this quantity is not positive definite in the tangent bundle . Nevertheless, it is possible to see that magnitudes of and remain small if at initial time both magnitudes and are set small. Next, let us introduce the equilibrium manifold defined by the setwhich is of one dimension. Fortunately as discussed in the paper , it is possible to confirm that a modified scalar functionbecomes positive definite in with an appropriate positive parameter , provided that the physical scale of the robot is ordinary, and feedback gains and are chosen adequately. Here, in (47), is defined as . Furthermore, if in a Riemannian ball in defined as the Jacobian matrix is nondegenerate, then it can be expected that there exist positive numbers and such thatFurther, to simplify the argument, we choose with a constant . Then, it follows from differentiation of thatwhereThis -vector is quadratic in components of and each coefficient of (for ) is at most of order of the maximum spectre radius of denoted by . Hence, it follows from (49) thatwhere signifies a constant that is of order of the maximum link length (because the Jacobian matrix is homogeneously related to the three link lengths of the robot shown in Figure 4). Further, note that in this case and remark thatThus, substituting (51) to (52) into (49) yieldsWith the aid of a positive parameter , it is easy to see that
Now, suppose that the robot has an ordinary physical scale withThen, it is possible to see thatprovided that [m], , and . From the practical point of view of control design for the handwriting robot, is set around and is chosen between (see ). On account of (48) and (56),and similarly,where . Then, if the damping factor can be selected to satisfy both inequalities and then (57) is reduced towhere can be set as which is larger than . Hence, by choosing , it follows from (60) and (59) thatwhere .
Now, suppose that in is the minimizing point that connects with among all geodesics from to any point of . We call a reference point corresponding to .
Definition 1 (stable Riemannian ball on a submanifold). If for any there exists
the number such that any
solution trajectory (orbit) of (43) starting from an arbitrary initial position
inside with remains inside for any , then the reference point on is said to be
stable on a submanifold (see Figure 5).
It can be concluded from the exponential convergence of to zero and the inequality when that any point inside included in for some is stable on a submanifold and further such a solution trajectory converges asymptotically to some on the equilibrium manifold 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 on the equilibrium manifold implies also the asymptotic convergence of constraint force to as because is expressed asand this right hand side converges to zero as .
The stability notion of a Riemannian ball in a neighborhood of a reference equilibrium state on is extended to cope with the case that the initial velocity vector is not zero. To do this, we define an extended Riemannian ball in the tangent bundle around in such a way thatwhere denotes the distance between and restricted to the submanifold and is defined below (40).
Definition 2 (asymptotic stability on a submanifold). If for any there exist numbers and such that any solution trajectory of (43) starting from an arbitrary initial position and velocity inside remains in and further converges asymptotically to some equilibrium point with still state, then the reference point with its posture on is said to be asymptotically stable on a constraint submanifold.
It should be remarked that Bloch et al.  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  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  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 -plane . We denote the object mass center by with the coordinates expressed in the inertial frame. On the other hand, we express a local coordinate system fixed at the object by - together with unit vectors and along the -axis and -axis, respectively (see Figure 7). The left-hand side surface of the object is expressed by a curve with local coordinates in terms of arc length parameter as shown in Figure 7.
First, suppose that at the left-hand contact point the fingertip of the left finger is contacting with the object. Denote the unit normal at by and the unit tangent vector by . Note that is normal to both the object surface and finger-end sphere at and is tangent to them at , too. If we denote position by local coordinates fixed at the object (see Figure 7), then the angle from the -axis to the unit normal is assumed to be determined by a function on the curve:where and . In this paper, all angles are set positive in counterclockwise direction. Then,which is dependent only on 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 constraintwhere denotes the right-hand side of (65) and for abbreviation.
Next, we note that the length can be regarded also as a shape function of the object given byOn the other hand, this quantity can be also expressed asAs discussed in , in the light of the book , pure rolling at is defined by the condition that the translational velocity of on the finger sphere is equal to that of on the object contour along the -axis. Hence, by denoting , we havewhich from (67) can be reduced toThis constraint form is Pfaffian. As to the contact point at the right-hand finger end, a similar nonholonomic constraint can be obtained. Thus, by introducing Lagrange's multipliers and associated with holonomic constraints , it is possible to construct a Lagrangian:where denote the joint vector for finger , the inertia matrix for finger , and 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 with another multipliers 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 , , and , andfor . It is important to note that the object dynamics of (73) and (74) can be recast in the formwhere andwhich 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 definethen (73) (75) can be written in the formwhereand signifies a function depending only on the shape parameter . 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 in (80), the image space of the Riemannian manifold with constraints and is spanned from the gradientsand the kernel space should be defined as the orthogonal compliment to the image space. Note that the rolling constraints of (71) and another for induce further restriction of the tangent space to a more subdimensional linear space. Here, should be taken at the time instant