7 The Dynamics of Systems of Interacting Rigid Bodies Kenneth A. Loparo Case Western Reserve University Ioannis S. Vakalis Institute for the Protection and Security of the Citizen (IPSC) European Commission 7.1 Introduction 7.2 Newton’s Law and the Covariant Derivative 7.3 Newton’s Law in a Constrained Space 7.4 Euler’s Equations and the Covariant Derivative 7.5 Example 1: Euler’s Equations for a Rigid Body 7.6 The Equations of Motion of a Rigid Body 7.7 Constraint Forces and Torques between Interacting Bodies 7.8 Example 2: Double Pendulum in the Plane 7.9 Including Forces from Friction and from Nonholonomic Constraints 7.10 Example 3: The Dynamics of the Interaction of a Disk and a Link 7.11 Example 4: Including Friction in the Dynamics 7.12 Conclusions 7.1 Introduction In this chapter, we begin by examining the dynamics of rigid bodies that interact with other moving or stationary rigid bodies. All the bodies are components of a multibody system and are allowed to have a single point of interaction that can be realized through contact or some type of joint constraint. The kinematics for the case of point contact has been formulated in previous works [52–54]. On the other hand, the case of joint constraints can be easily handled because the type of joint clearly defines the de- grees of freedom that are allowed for the rigid bodies that are connected through the joint. Then we will introduce a methodology for the description of the dynamics of a rigid body generally constrained by points of interaction. Our approach is to use the geometric properties of Newton’s equations and Euler’s equations to accomplish this objective. The methodology is developed in two parts, we first investigate the geometric properties of the basic equations of motion of a rigid body. Next we consider a multibody system that includes point interaction that can occur through contact or some type of joint constraint. Each body is considered initially as an independent unit and forces and torques are applied to the bodies through the interaction points. There is a classification of the applied forces with respect to their type Copyright © 2005 by CRC Press LLC The Dynamics of Systems of Interacting Rigid Bodies 7 -3 through an immersion. This requires a positive definite Riemannian metric on the first manifold and the extension to a pseudo-Riemannian metric requires further investigation and will not be discussed here. In order to establish a geometric form for the acceleration component of Newton’slaw,weneedtoin- troduce the notion of a connection on a general Riemannian manifold. The connection is used to describe the acceleration along curves in more general spaces like a Riemannian manifold, see Boothby [12]. Definition 7.2 A C ∞ connection ∇ on a manifold N is a mapping ∇ : X(N) × X(N) −→ X(N) defined by ∇(X, Y) −→ ∇ X Y,∇ satisfies the linearity properties for all C ∞ functions f, g on N and X, X , Y, Y ∈ X(N): 1. ∇ fX+gX Y = f (∇ X Y) + g(∇ X Y) 2. ∇ X ( fY+ gY ) = f ∇ X Y + g∇ X Y + (Xf)Y + (Xg)Y Here, X(M) denotes the set of C ∞ vector fields on the manifold M. Definition 7.3 A connection on a Riemannian manifold N is called a Riemannian connection if it has the additional properties: 1. [X, Y] =∇ X Y −∇ Y X 2. XY, Y =∇ X Y, Y +Y, ∇ X Y Here, [·, ·] denotes the Lie bracket. Theorem 7.1 If N is a Riemannian manifold, then there exists a uniquely determined Riemannian con- nection on N. A comprehensive proof of this theorem can be found in W.M. Boothby [12]. The acceleration along a curve c(t) ∈ N is given by the connection ∇ ˙ c(t) ˙ c(t). In this context the Riemannian connection denotes the derivative of a vector field along the direction of another vector field, at a point m ∈ N. To understand how the covariant derivative, defined on a submanifold M of IR n , leads to the abstract notion of a connection, we need to introduce the derivative of a vector field along a curve in IR n . Consider a vector field X definedonIR n and a curve c(t) ∈ IR n .LetX(t) = X | c(t) , then the derivative of X(t) denoted by ˙ X(t) is the rate of change of the vector field X along this curve. Consider a point c(t 0 ) = p ∈ IR n and the vectors X(t 0 ) ∈ T c(t 0 ) IR n and X(t 0 + t) ∈ T c(t 0 +t) IR n . We can use the natural identification of T c(t 0 ) IR n with T c(t 0 +t) IR n , and the difference X(t 0 +t) −X(t 0 ) can be defined in T c(t 0 ) IR n . Consequently, the derivative ˙ X(t 0 ) can be defined by ˙ X(t 0 ) = lim t→0 X(t 0 + t) − X(t 0 ) t Consider a submanifold M imbedded in IR n , and a vector field X on M, not necessarily tangent to M. Then, the derivative of X along a curve c(t) ∈ M is denoted by ˙ X(t) ∈ T c(t) IR n .Atapointc(t 0 ) = p ∈ M, the tangent space T p IR n can be decomposed into two mutually exclusive subspaces T p IR n = T p M ⊕T p M ⊥ . Consider the projection p 1 : T p IR n −→ T p M, the covariant derivative of X along c(t) ∈ M is defined as follows: Definition 7.4 The covariant derivative of a vector field X on a submanifold M of IR n along a curve c(t) ∈ M is the projection p 1 ( ˙ X(t)) and is denoted by DX dt . An illustration of the covariant derivative is given in Figure 7.1. The covariant derivative gives rise to the notion of a connection through the following construction: Consider a curve c(t) ∈ M, the point p = c(t 0 ),andthetangentvectortothe curve X p = ˙ c(t 0 )atp. Wecandefinethemap∇ X p : T p M −→ T p M by ∇ X p Y : X p −→ DY dt | t=t 0 , along any curve c(t) ∈ M such that ˙ c(t 0 ) = X p and Y ∈ M. Along the curve c(t)wehave∇ ˙ c(t) Y = DY dt . The connection can be defined asa map ∇ : X(M)×X(M) −→ X(M), where Copyright © 2005 by CRC Press LLC 7 -6 Robotics and Automation Handbook When studying the dynamicsof rigid bodies,we deal with Riemannian metrics, whichare tensors oftype T 0 2 (N ). Using the general theorem for pulling back covariant tensors and the definition of an immersion, the following theorem results, see F. Brickell and R.S. Clark [13]: Theorem 7.3 If a manifold N has a given positive definite Riemannian metric then any global immersion f : N −→ N induces a positive definite Riemannian metric on N. A manifold N is a submanifold of a manifold N if N is a subset of N and the natural injection i : N −→ N is an immersion. Submanifolds of this type are called immersed submanifolds. If in addition, we restrict the natural injection to be one to one, then we have an imbedded submanifold [13]. There is an alternative definition for a submanifold where the subset property is not included along with the C ∞ structure, see W.M. Boothby [12]. In our case, because we are dealing with the constrained configuration space of a rigid body which is subset of IE(3), it seems more natural to follow the definition of a submanifold given by F. Brickell and R.S. Clark [13]. From the previous theorem we conclude that we can induce a Riemannian metric on a manifold N if there is a Riemannian manifold N and a map f : N −→ N , which is an immersion. The manifold N need not be a submanifold of N . In general, N will be a subset of N and the map f : N −→ N is needed to induce a metric. We consider a system of rigid bodies with each body of the system as a separate unit moving in a constraint state space because of the presence of some joint or point contact beween it and the rest of the bodies. The case of a joint constraint is easier to deal with because the type of joint also defines the degrees of freedom of the constrained body. The degrees of freedom then define the map that describes the constrained configuration space of motion as a submanifold of IE(3). Accordingly we can then induce a Riemannian metric on the constrained configuration space. The case of point contact is more complicated. We have studied the problem of point contact, where the body moves on a smooth surface B in [53]. In this work we have shown that the resulting constrained configuration space M is a subset of IE(3). M is also a submanifold of IE(3) × B and there is a map µ 1 : M −→ IE(3). This map is not necessarily the natural injection i : M −→ IE(3). The map µ 1 therefore should be an immersion so that it induces a Riemannian metric on M from IE(3). This is not generally the case when we deal with the dynamics of constrained rigid body motions defined by point contact. Using this analysis we can now study the geometric form of Newton’s law for bodies involved in constrained motion. To begin with we need to endow the submanifold M of IR 3 with a Riemannian structure. According to the discussion above this is possible using the pull-back of the Riemannian metric σ on IR 3 through the natural injection j : M −→ IR 3 ,wherej is by definition an immersion. We let ¯ σ = j ∗ σ denote the induced Riemannian structure on the submanifold M by the pull-back j ∗ . We can explicitly compute the coordinate representation of the pull-back of the Riemannian metric ¯ σ if we use the fact that positive definite tensors of order T 0 2 (N ) on a manifold N can be represented as positive definite symmetric matrices. In our case, the Riemannian metric in IR 3 can be represented as a 3 × 3 positive definite symmetric matrix given by σ = σ 11 σ 12 σ 13 σ 12 σ 22 σ 23 σ 13 σ 23 σ 33 Assume that the submanifold M is two-dimensional and has coordinates {x 1 , x 2 }, and {y 1 , y 2 , y 3 } are the coordinates of IR 3 . Then, the derived map j ∗ : TM −→ TIR 3 has the coordinate representation: j ∗ = ∂ j 1 ∂x 1 ∂ j 1 ∂x 2 ∂ j 2 ∂x 1 ∂ j 2 ∂x 2 ∂ j 3 ∂x 1 ∂ j 3 ∂x 2 Copyright © 2005 by CRC Press LLC The Dynamics of Systems of Interacting Rigid Bodies 7 -7 The induced Riemannian metric ¯ σ on M (by using the pull-back) has the following coordinate representation: ¯ σ = j ∗ σ = ∂ j 1 ∂x 1 ∂ j 2 ∂x 1 ∂ j 3 ∂x 1 ∂ j 1 ∂x 2 ∂ j 2 ∂x 2 ∂ j 3 ∂x 2 · σ 11 σ 12 σ 13 σ 12 σ 22 σ 23 σ 13 σ 23 σ 33 · ∂ j 1 ∂x 1 ∂ j 1 ∂x 2 ∂ j 2 ∂x 1 ∂ j 2 ∂x 2 ∂ j 3 ∂x 1 ∂ j 3 ∂x 2 In a similar way we can pull-back 1-forms from T ∗ N to T ∗ N, considering that 1-forms are actually tensors of order T 0 1 (N ). We can represent a 1-form ω in the manifold N , with coordinates {y 1 , , y n }, as ω = a 1 dy 1 +···+a n dy n . Then using the general form for a tensor of type T 0 r (N ), the pull-back of ω denoted by ¯ ω = f ∗ ω, has the coordinate representation: ¯ ω = ∂ f 1 ∂x 1 ··· ∂ f n ∂x 1 . . . . . . ∂ f 1 ∂x m ··· ∂ f n ∂x m a 1 . . . a n where f = f 1 , , f n is the coordinate representation of f and {x 1 , , x m } are the coordinates of N. Using this formula we can “pull-back” the 1-forms that represent the forces acting on the object. Thus on the submanifold M, the forces acting on the object are described by ¯ F = j ∗ F . If, for example, we consider the case where M is a two-dimensional submanifold of IR 3 , then the 1-form representing the forces applied on the object F = F 1 dy 1 + F 2 dy 2 + F 3 dy 3 is pulled back to ¯ F = j ∗ F = ∂ j 1 ∂x 1 ∂ j 2 ∂x 1 ∂ j 3 ∂x 1 ∂ j 1 ∂x 2 ∂ j 2 ∂x 2 ∂ j 3 ∂x 2 · F 1 F 2 F 3 Next we use the results from tensor analysis and Newton’s law to obtain the dynamic equations of a rigid body on the submanifold M.InNewton’s law the flat map σ , resulting from the Riemannian metric σ , is used. A complete description of the various relations is given by the commutative diagram TIK 2 T*IK 2 TM T*m σ b σ b j * j* Let ¯ c = c | M denote the restriction of the curve c(t)toM.Let ¯ ∇ denote the connection on M associated with the Riemannian metric ¯ σ . Then we can describe Newton’slawonM by ¯ σ ( ¯ ∇ ˙ ¯ c(t) ˙ ¯ c(t)) = ¯ F (7.2) We have to be careful how we interpret the different quantities involved in the constrained dynamic equation above, because the notation can cause some confusion. Actually, the way we have written Copyright © 2005 by CRC Press LLC The Dynamics of Systems of Interacting Rigid Bodies 7 -9 G g(t) e Φ(1, e) T e G v FIGURE 7.2 The action of the exponential map on a group. Where (1, e) is the result of the flowofavectorfield X applied on the identity element e for t = 1. Figure 7.2 illustrates how the exponential map operates on elements of a group G. Consider the tangent space of the Lie group G at the identity element e, denoted by T e G. The Lie algebra of the Lie group G is identified with T e G, and it can be denoted by V. The main idea underlying this analysis is the fact that we can identify a chart in V, in the neighborhood of 0 ∈ V, with a chart in the neighborhood of e ∈ G.Morespecifically, what V.I. Arnold proves in his work is that Euler’s equations for a rigid body are equations that describe geodesic curves in the group SO(3), but they are described in terms of coordinates of the corresponding Lie algebra so(3). This is done by the identification mentioned above using the exponential map exp : so(3) −→ SO(3). We are going to outline the main concepts and theorems involved in this construction. A complete presentation of the subject is included in [5]. A Lie group acts on itself by left and right translations. Thus for every element g ∈ G, we have the following diffeomorphisms: L g : G −→ G, L g h = gh R g : G −→ G, R g h = hg As a result, we also have the following maps on the tangent spaces: L g∗ : T g G −→ T gh G, R g∗ : T g G −→ T gh G Next we consider the map R g −1 L g : G −→ G, which is a diffeomorphism on the group. Actually, it is an automorphism because it leaves the identity element of the group fixed. The derived map of R g −1 L g is going to be very useful in the construction which follows: Ad g = (R g −1 L g ) e∗ : T e G = V −→ T e G = V Thus, Ad g is a linear map of the Lie algebra V to itself. The map Ad g has certain properties related to the Lie bracket [·, ·] of the Lie algebra V. Thus, if exp : V −→ G and g(t) = exp( f (t)) is a curve on G, then we have the following relations: Ad exp( f (t)) ξ = ξ + t[ f, ξ] +o(t 2 )(t → 0) Ad g [ξ, n] = [Ad g ξ, Ad g n] There are two linear maps induced by the left and right translations on the cotangent space of G, T ∗ G. These maps are the duals to L g∗ and R g∗ and are known as pull-backs of L g and R g , respectively: L ∗ g : T ∗ gh G −→ T ∗ h G, R ∗ g : T ∗ hg G −→ T ∗ h G Copyright © 2005 by CRC Press LLC 7 -10 Robotics and Automation Handbook We have the following properties for the dual maps: (L ∗ g ξ, n) = (ξ, L g∗ n) (R ∗ g ξ, n) = (ξ, R g∗ n) for ξ ∈ T ∗ g G, n ∈ T g G. Here, (ξ, n) ∈ IR is the value of the linear map ξ applied on the vector field n, both evaluated at g ∈ G. In general we can define a Euclidean structure on the Lie algebra V via a symmetric positive definite linear operator A : V −→ V ∗ (A : T e G −→ T ∗ e G), (Aξ, n) = (An, ξ) for all ξ, n ∈ V. Using theleft translation we can define asymmetric operator A g : T g G −→ T ∗ g G according to the relation A g ξ = L ∗ g −1 AL g −1 ∗ ξ. Finally, via this symmetric operator we can defineametriconT g G by ξ, n g = (A g n, ξ) = (A g ξ, n) =n, ξ g for all ξ, n ∈ T g G. The metric ·, · g is a Riemannian metric that is invariant under left translations. At the identity element e, we denote the metric on T e G = V by ·, ·.Wecandefine an operator B : V ×V −→ V using the relation [a, b], c=B(c, a), b for all b ∈ V. B is a bilinear operator, and if we fix the first argument, B is skew symmetric with respect to the second argument: B(c, a), b+B(c, b), a=0 In the case of rigid body motions the group is SO(3) and the Lie algebra is so(3), but we are going to keep the general notation in order to emphasize the fact that this construction is more general and can be applied to a variety of problems. The rotational part of the motion of a body can be represented by a trajectory g (t) ∈ G.Thus ˙ g represents the velocity along the trajectory ˙ g ∈ T g(t) G. The rotational velocity with respect to the body coordinate frame is the left translation of the vector ˙ g ∈ G to T e G = V. Thus, if we denote the rotational velocity of the body with respect to the coordinate frame attached to the body by ω c , then we have ω c = L g −1 ∗ ˙ g ∈ V. In a similar manner the rotational velocity of the body with respect to an inertial (stationary) frame is the right translation of the vector ˙ g ∈ T g(t) G to T e G = V, which we denote by ω s = R g −1 ∗ ˙ g ∈ V. In this case we have that A g : T g G −→ T ∗ g G is an inertia operator. Next, we denote the inertia operator at the identity by A : T e G −→ T ∗ e G. The angular momentum M = A g ˙ g ∈ V can be expressed with respect to the body coordinate frame as M c = L ∗ g M = Aω c and with respect to an inertial frame M s = R ∗ g M = Ad ∗ g −1 M (Ad ∗ g −1 is the dual of Ad g −1 ∗ ). Euler’s equations according to the notation established above are given by dω c dt = B(ω c , ω c ), ω c = L g −1 ∗ ˙ g This form of Euler’s equations can be derived in two steps. Consider first a geodesic g (t) ∈ G such that g(0) = e and ˙ g(0) = ω c . Because the metric is left invariant, the left translation of a geodesic is also a geodesic. Thus the derivative dω c dt depends only on ω c and not on g. Using the exponential map, we can consider a neighborhood of 0 ∈ V as a chart of a neighborhood of the identity element e ∈ G. As a consequence, the tangent space at a point a ∈ V, namely T a V, is identified naturally with V. Thus the following lemma can be stated. Lemma 7.1 Consider the left translation L exp(a) for a ∈ V. This map can be identified with L a for a→0. The corresponding derived map is denoted by L ∗a , and L ∗a : V = T 0 V → V = T a V.Ifξ ∈ V then L a∗ ξ = ξ + 1 2 [a, ξ] + o(a 2 ) Because geodesics can be translated to the origin using coordinates of the algebra V, the derivative dω c dt gives the Euler equations. The proof of the lemma as well as more details on the rest of the arguments can be found in [5]. Copyright © 2005 by CRC Press LLC 7 -12 Robotics and Automation Handbook With respect to an inertial coordinate frame, the velocity of the body is ω s = R g −1 ∗ ˙ g, which when given in matrix form yields ω s1 ω s2 ω s3 = 0 −sin(φ) sin(θ) cos(φ) 0 cos(φ) sin(θ) sin(φ) 1 0 cos(θ) ˙ φ ˙ θ ˙ ψ The kinetic energy metric is given by K = 1 2 ˙ g, ˙ g, g = 1 2 ω c , ω c . We can choose a coordinate frame attached to the body such that the three axes are the principle axes of the body; thus, A = I 1 00 0 I 2 0 00I 3 We can calculate the quantity A g = L ∗ g −1 AL g −1 ∗ , which when given in matrix form yields A g = −sin(θ) cos(ψ) sin(θ) sin(ψ) cos(θ) sin(ψ) cos(ψ)0 001 I 1 00 0 I 2 0 00I 3 −sin(θ) cos(ψ) sin(ψ)0 sin(θ) sin(ψ) cos(ψ)0 cos(θ)01 We can write Euler’s equations in group coordinates using the formula ρ (∇ ˙ g ˙ g) = T The complete system of Euler’s equations, using the coordinates given by the three Euler angles, is given by ¨ φ I 2 sin(ψ) 2 sin(θ) 2 + 2 ˙ φ ˙ ψ I 2 cos(ψ) sin(ψ) sin(θ) 2 − 2 ˙ φ ˙ ψ I 1 cos(ψ) sin(ψ) sin(θ) 2 + ¨ φ I 1 cos(ψ) 2 sin(θ) 2 +2 ˙ φ ˙ θ I 2 sin(ψ) 2 cos(θ) sin(θ) +2 ˙ φ ˙ θ I 1 cos(ψ) 2 cos(θ) sin(θ) −2 ˙ φ ˙ θ I 3 cos(θ) sin(θ) − ˙ ψ ˙ θ I 2 sin(ψ) 2 sin(θ) + ˙ ψ ˙ θ I 1 sin(ψ) 2 sin(θ) + ¨ θ I 2 cos(ψ) sin(ψ) sin(θ) − ¨ θ I 1 cos(ψ) sin(ψ) sin(θ) + ˙ ψ ˙ θ I 2 cos(ψ) 2 sin(θ) − ˙ ψ ˙ θ I 1 cos(ψ) 2 sin(θ) − ˙ ψ ˙ θ I 3 sin(θ) + ¨ φ I 3 cos(θ) 2 + ˙ θ 2 I 2 cos(ψ) sin(ψ) cos(θ) − ˙ θ 2 I 1 cos(ψ) sin(ψ) cos(θ) + ¨ ψ I 3 cos(θ) = T φ − ˙ φ 2 I 2 sin(ψ) 2 cos(θ) sin(θ) − ˙ φ 2 I 1 cos(ψ) 2 cos(θ) sin(θ) + ˙ φ 2 I 3 cos(θ) sin(θ) − ˙ φ ˙ ψ I 2 sin(ψ) 2 sin(θ) + ˙ φ ˙ ψ I 1 sin(ψ) 2 sin(θ) + ¨ φ I 2 cos(ψ) sin(ψ) sin(θ) − ¨ φ I 1 cos(ψ) sin(ψ) sin(θ) + ˙ φ ˙ ψ I 2 cos(ψ) 2 sin(θ) − ˙ φ ˙ ψ I 1 cos(ψ) 2 sin(θ) + ˙ φ ˙ ψ I 3 sin(θ) + ¨ θ I 1 sin(ψ) 2 − 2 ˙ ψ ˙ θ I 2 cos(ψ) sin(ψ) +2 ˙ ψ ˙ θ I 1 cos(ψ) sin(ψ) + ¨ θ I 2 cos(ψ) 2 = T θ − ˙ φ 2 I 2 cos(ψ) sin(ψ) sin(θ) 2 + ˙ φ 2 I 1 cos(ψ) sin(ψ) sin(θ) 2 + ˙ φ ˙ θ I 2 sin(ψ) 2 sin(θ) − ˙ φ ˙ θ I 1 sin(ψ) 2 sin(θ) − ˙ φ ˙ θ I 2 cos(ψ) 2 sin(θ) + ˙ φ ˙ θ I 1 cos(ψ) 2 sin(θ) − ˙ φ ˙ θ I 3 sin(θ) + ¨ φ I 3 cos(θ) + ˙ θ 2 I 2 cos(ψ) sin(ψ) − ˙ θ 2 I 1 cos(ψ) sin(ψ) + ¨ ψ I 3 = T ψ Observe that the torques T are given in terms of the axes of the Euler angles and, thus, cannot be measured directly. This happens because the axis of rotation for an Euler angle is moving independent of the body during the motion. This can be circumvented by using the torques around the principle axes, and then expressing them via an appropriate transformation to torques measured with respect to the axes Copyright © 2005 by CRC Press LLC 7 -14 Robotics and Automation Handbook we can pull-back the Riemannian metric on M from IE(3) using the natural injection from M to IE(3), whichisbydefinition an immersion. Actually this is the case that occurs when the rigid bodies interact through a joint. Depending on the type of joint, we can create a submanifold of IE(3) by considering only the degrees of freedom allowed by the joint. The other case that we might encounter is when M is a subset of IE(3), which is also a manifold (with differential structure established independent of the fact that is a subset ) and there is a map l : M −→ IE(3). This is what happens in the case where the rigid bodies interact with contact. As we proved in [53], the subset M is also a manifold with a differential structure established by the fact that M is a submanifold of IE(3) ×B. Also we proved in [53] that there exists a map µ 1 : M −→ IE(3). In the case that this map is an immersion, we can describe the dynamics of the rigid body on M. According to our assumptions about the point interaction between rigid bodies, joints and contact are the two types of interactions that are being considered. Our intention is to develop a general model for the dynamics of a set of rigid interacting bodies; therefore, both cases of contact and joint interaction might appear simultaneously in a system of equations describing the dynamics of each rigid body in the rigid body system. Thus we will consider the more general case where the dynamics are constrained to a subset M of IE(3), which isa manifold, and that there exists a mapl : M −→ IE(3) which is an immersion. We can use as before the pull-back l ∗ : T ∗ IE(3) −→ T ∗ M to induce a metric on M. If we denote the Riemannian metric on IE(3) by = σ +ρ, then the induced Riemannian metric on M is ¯ = l ∗ . Consider next a trajectory on IE(3), r (t):(c(t), g(t)). If this curve is a solution for the dynamics of a rigid body, it satisfies Equations (7.3) which can be written in condensed form (∇ ˙ r (t) ˙ r (t)) = F + T (7.4) The term ∇ ˙ r (t) ˙ r (t) has the same form as the RHS of the geodesic equations and depends only on the corresponding Riemannian metric . When we describe the dynamics on the manifold M, with the induced Riemannian metric ¯ , the corresponding curve ¯ r (t) ∈ M should satisfy the following equations: ¯ ( ¯ ∇ ˙ ¯ r (t) ˙ ¯ r (t)) = l ∗ (F + T) (7.5) where F +T is the combined 1-form of the forces and the torques applied to the unconstrained body. The term ¯ ∇ ˙ ¯ r (t) ˙ ¯ r (t) is again the same as the RHS of the equations for a geodesic curve on M and depends only on the induced Riemannian metric ¯ . Thus in general, the resulting curve ¯ r (t) ∈ M will not be related to the solution of the unconstrained dynamics except in the case where M is a submanifold of IE(3), and in this case we have ¯ r (t) = r(t) | M . It is natural to conclude that the 1-form l ∗ (F + T) includes all forces and torques that allow the rigid body to move on the constraint manifold M. In the initial configuration space IE(3), the RHS of the dynamic equations includes all the forces and torques that make the body move on the subset M of IE(3); these are sometimes referred to as “generalized constraint forces.” Let F M and T M denote, respectively, these forces and torques. We can separate the forces and torques into two mutually exclusive sets, one set consisting of the forces and torques that constrain the rigid body motion on M and the other set of the remaining forces and torques which we denote by F S , T S .Thuswehave F + T = (F M + T M ) ⊕ (F S + T S ). We can define a priori the generalized constraint forces as those that satisfy the condition l ∗ (F M + T M ) = 0 (7.6) This is actually what B. Hoffmann described in his work [20] as Kron’s method of subspaces. Indeed in this work Kron’s ideasabout the use oftensor mathematics in circuit analysisare appliedto mechanical systems. As Hoffman writes, Kron himself gave some examples of the application of his method in mechanical systems in an unpublished manuscript. In this chapter we actually develop a methodology within Kron’s framework, using the covariant derivative, for the modeling of multibody systems. From the analysis that we have presented thus far it appears as if this is necessary in order to include not only joints between the bodies of the system but also point contact interaction. Copyright © 2005 by CRC Press LLC The Dynamics of Systems of Interacting Rigid Bodies 7 -17 The degrees of freedom of the system are θ, φ, x and y, and the matrix expression for ∇ ˙ r (t) ˙ r (t)is ¨ θ ¨ φ ¨ x ¨ y The matrix expression of the 1-form of the forces and the torques F + T is F + T = −m 1 l 1 g sin(θ ) − F x l 1 cos(θ) + F y l 1 sin(θ) −F x l 2 cos(φ) + F y l 2 sin(φ) F x −F y − m 2 g At this point we have described the equations of motion for the two free bodies that make up the system. Next we proceed with the reduction of the model according to the methodology described above. The degrees of freedom of the constrained system are ˜ θ, ˜ φ, and we can construct the map ξ according to ξ : { ˜ θ, ˜ φ}−→{ ˜ θ, ˜ φ, l 1 sin( ˜ θ) +l 2 sin( ˜ φ), −l 1 cos( ˜ θ) −l 2 cos( ˜ φ)} Next ξ ∗ is calculated: ξ ∗ = 10 01 l 1 cos( ˜ θ) l 2 cos( ˜ φ) l 1 sin( ˜ θ) l 2 sin( ˜ φ) It is obvious that the map ξ is an immersion, because the rank(ξ ∗ ) = 2. Consequently, we compute ¯ ρ = ξ ∗ ρ: ¯ ρ = l 2 1 m 1 +l 2 1 m 2 l 1 l 2 m 2 cos( ˜ θ − ˜ φ) l 1 l 2 m 2 cos( ˜ θ − ˜ φ) l 2 2 m 2 Applying ξ ∗ on the 1-forms of the forces and the torques on the RHS of the dynamic equations of motion we obtain ξ ∗ (F + T) = −m 1 l 1 gsin( ˜ θ) −l 1 m 2 gsin( ˜ θ) −l 2 m 2 gsin( ˜ φ) We observe that all the reaction forces have disappeared from the model because we are on the constrained submanifold. Thus, we could have neglected them in the initial free body modeling scheme. We continue with the calculation of ¯ ∇ ˙ ¯ r (t) ˙ ¯ r (t), which defines a geodesic on the constraint manifold. The general form of the geodesic is [12,14] d 2 x i dt 2 + i jk dx j dt dx k dt Copyright © 2005 by CRC Press LLC 7 -20 Robotics and Automation Handbook (y 1 , y 2 ) (z 1 , z 2 ) v –v B u t 1 FIGURE 7.5 Disk in contact with a link. assume that there are no friction forces generated by the contact. The set of the dynamic equations is m ¨ z 1 =−vsin(θ 2 ) m ¨ z 2 = u − vcos(θ 2 ) −mg I ¨ θ = 0 m 2 ¨ y 1 = vsin(θ 2 ) m 2 ¨ y 2 = vcos(θ 2 ) −m 2 g I 2 ¨ θ 2 = u 2 − vτ 1 Where m, I are the mass and the moment of inertia of the disk, m 2 , I 2 are the mass and the moment of inertia of the link, and u 2 is the input (applied) torque. As we mentioned previously, the dynamic equations that describe the motion for each body are described initially on IE(2). The torques and the forces are 1-forms in T ∗ IE(2). The dynamic equations of the combined system are defined on IE(2) ×IE(2), but the constrained system actually evolves on M 1 × M 2 . We can restrict the original system of dynamic equations to M 1 × M 2 using the projection method developed earlier. Thus we need to construct the projection function ξ : M 1 × M 2 −→ IE(2) × IE(2). We already have a description of the projection mappings for each constrained submanifold, µ 1 : M 1 −→ IE(2) and ν 1 : M 2 −→ IE(2). We can use the combined projection function ξ = (µ 1 , ν 1 ) for the projection method. As stated previously, the forces and torques are 1-forms in T ∗ IE(2), and for the link, for example, they should have the form a(y 1 , y 2 , θ 2 )dy 1 +b(y 1 , y 2 , θ 2 )dy 2 +c(y 1 , y 2 , θ 2 )dθ 2 . However, in the equationsfor the 1-forms we have the variables τ 1 and u 2 , which are different from {y 1 , y 2 , θ 2 }. The variable u 2 is the input (torque) to the link, and τ 1 is the point of interaction between the link and the disk. Thus τ 1 can be viewed as another input to the link system. In other words, when we have contact between objects, the surfaces of the objects in contact must be included in the input space. When we project the dynamics on the constraint submanifold M 2 , then τ 1 is assigned a special value because in M 2 , τ 1 is a function of { ˜ y 1 , ˜ θ 2 } (the coordinates of M 2 ). We can compute the quantities ξ ∗ and ξ ∗ that are needed for the modeling process. The projection map ξ is defined as ξ : { ˜ z 1 , ˜ θ, ˜ y 1 , ˜ θ 2 }−→ ˜ z 1 ,1, ˜ θ, ˜ y 1 , 1 +sin(θ 2 )( ˜ z 1 − ˜ y 1 ) cos( ˜ θ 2 ) + 1, ˜ θ 2 Copyright © 2005 by CRC Press LLC [...]... Robotics and Automation, 218 5 218 9 Copyright © 20 05 by CRC Press LLC 7-2 4 Robotics and Automation Handbook [27] Krishnaprasad, P S., Yang, R., and Dayawansa, W.P (19 91) Control problems on principle bundles and nonholonomic mechanics Proceedings of the 30th IEEE Conference on Decision and Control, 11 33 11 37 [28] Latombe, J.-C (19 91) Robot Motion Planning Kluwer, Boston [29] Laumond, J.-P (19 93) Controllability... Conference on Decision and Control, 2944–2949 [43] Samson, C (19 91) Velocity and torque feedback control of a nonholonomic cart Advanced Robot Control, C Canudas de Wit, ed., LNCIS 16 2, Springer-Verlag, Germany, 12 5 15 1 [44] Samson, C (19 93) Time-varying feedback stabilization of a car-like wheeled mobile robot Int J Robotics Res., 12 , 1, 55 –66 [ 45] Samson, C and Ait-Abderrahim, K 19 91 Feedback stabilization... matrix A M Equation (8 .12 ) is the same as Equation (8 .11 ) except that the point P F has been written Copyright © 20 05 by CRC Press LLC 8-4 Robotics and Automation Handbook link i + 1 Xi axis i + 1 zi {Fi} ai Oi di = link offset link i ai = link length about xi a i = Zi 1 Zi about zi 1 q i = Xi 1 Xi ai Xi 1 {Fi 1} axis i Oi 1 O′ Zi 1 qi link i − 1 di FIGURE 8.3 Schematic of the adjacent axes with the... Controllability of a multibody mobile robot IEEE Transactions on Robotics and Automation, 9, 6, 755 –763 [30] Laumond, J.-P., Jacobs, P.E., Taix, M., and Murray, R.M (19 94) A motion planner for nonholonomic systems IEEE Transactions on Robotics and Automation, 10 , 5, 57 7 59 3 [ 31] Laumond, J.-P., Sekhavat, S., and Vaisset, M (19 94) Collision-free motion planning for a nonholonomic mobile robot with trailers... reorienting linked rigid bodies using internal motions IEEE Trans Robotics Automation, 11 , 1, 13 9 14 5 Copyright © 20 05 by CRC Press LLC 8 D-H Convention 8 .1 8.2 8.3 Dr Jaydev P Desai Drexel University 8 .1 8.4 Introduction D-H Parameters Algorithm for Determining the Homogenous Transformation Matrix, A◦ n Examples Introduction Denavit-Hartenberg in 19 55 developed a notation for assigning orthonormal coordinate... 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Copyright © 20 05 by CRC Press LLC 7 -2 4 Robotics and Automation Handbook [27]. written Copyright © 20 05 by CRC Press LLC 8 -4 Robotics and Automation Handbook X i d i a i O i z i link i + 1 link i − 1 link i axis i + 1 axis i {F i } {F i 1 } q i a i O i 1 Z i 1 O′ X i 1 d i = link