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
Trang 17 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
Trang 2The 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’s law, we need to
in-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 ∇f X +g XY = f (∇X Y ) + g(∇XY )
2 ∇X( f Y + gY)= f ∇X Y + g∇X Y+ (X f )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 ]= ∇XY− ∇YX
2 XY, Y = ∇XY, 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 IRn Consider a
vector field X defined on IR n and a curve c (t)∈ IRn Let X(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 (t0) = p ∈ IR n and the vectors X(t0)∈ Tc (t0 )IRn and X(t0+ t) ∈ Tc (t0+t)IRn We can use the natural identification of
T c (t0)IRn with Tc (t0+t)IRn , and the difference X(t0+ t) − X(t0) can be defined in T c (t0)IRn Consequently, the derivative ˙X(t0) can be defined by
˙
X(t0)= lim
t→0
X(t0+ t) − X(t0)
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) ∈ Tc (t)IR n At a point c (t0)= p ∈ M, the tangent space TpIR n can be decomposed into two mutually exclusive subspaces TpIR n = Tp M ⊕Tp M⊥
Consider the projection p1: TpIR n −→ Tp 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 nalong a curve
c (t) ∈ M is the projection p1( ˙X(t)) and is denoted by D X dt
An illustration of the covariant derivative is given inFigure 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(t0), and the tangent vector to the curve X p = ˙c(t0) at p We can define the map∇Xp : T p M −→ Tp M
by∇Xp Y : X p−→ DY
dt |t=t0, along any curve c (t) ∈ M such that ˙c(t0)= X p and Y ∈ M Along the curve
c (t) we have∇˙c (t) Y = DY
dt The connection can be defined as a map∇ : X (M)×X (M) −→ X (M), where
Trang 37-6 Robotics and Automation Handbook
When studying the dynamics of rigid bodies, we deal with Riemannian metrics, which are tensors of type
T0(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 Nhas a given positive definite Riemannian metric then any global immersion
f : N −→ Ninduces a positive definite Riemannian metric on N.
A manifold N is a submanifold of a manifold N if N is a subset of Nand 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 Nand 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 Nand the map f : N −→ Nis
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 µ1 : M−→ IE(3) This map is
not necessarily the natural injection i : M −→ IE(3) The map µ1therefore 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 IR3 with a Riemannian structure According to the discussion above this is possible using the pull-back of the Riemannian metric
σ on IR3through the natural injection j : M −→ IR3, where j 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 T0(N) on a manifold Ncan be represented as positive definite symmetric matrices In our case, the Riemannian metric in IR3can be represented as a 3× 3 positive definite symmetric matrix given by
σ =
σ σ11 σ12 σ13
12 σ22 σ23
σ13 σ23 σ33
Assume that the submanifold M is two-dimensional and has coordinates {x1, x2}, and {y1, y2, y3} are the coordinates of IR3 Then, the derived map j∗: TM −→ TIR3has the coordinate representation:
j∗=
∂ j1
∂x1 ∂ j1
∂x2
∂ j2
∂x1 ∂ j2
∂x2
∂ j3
∂x1 ∂ j3
∂x2
Copyright © 2005 by CRC Press LLC
Trang 4The 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 =
∂ j1
∂x1 ∂ j2
∂x1 ∂ j3
∂x1
∂ j1
∂x2 ∂ j2
∂x2 ∂ j3
∂x2
·
σ11 σ12 σ13
σ12 σ22 σ23
σ13 σ23 σ33
·
∂ j1
∂x1 ∂ j1
∂x2
∂ j2
∂x1 ∂ j2
∂x2
∂ j3
∂x1 ∂ j3
∂x2
In a similar way we can pull-back 1-forms from T∗Nto T∗N, considering that 1-forms are actually
tensors of order T0(N) We can represent a 1-formω in the manifold N, with coordinates{y1, , y n},
asω = a1d y1+ · · · + an d y n Then using the general form for a tensor of type T0
r (N), the pull-back ofω
denoted by ¯ω = f∗ω, has the coordinate representation:
¯
ω =
∂ f1
∂x1 · · · ∂ f n
∂x1
∂ f1
∂x m · · · ∂ f n
∂x m
a1
a n
where f = f1, , f n is the coordinate representation of f and {x1, , 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 IR3, then the 1-form representing the forces applied
on the object F = F1d y1+ F2d y2+ F3d y3is pulled back to
¯
F = j∗F =
∂ j1
∂x1 ∂ j2
∂x1 ∂ j3
∂x1
∂ j1
∂x2 ∂ j2
∂x2 ∂ j3
∂x2
·
F1
F2
F3
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 In Newton’s law the flat map σ , resulting from the Riemannian metricσ,
is used A complete description of the various relations is given by the commutative diagram
σ b
σ b
j
Let ¯c = c |M denote the restriction of the curve c (t) to M Let ¯ ∇ denote the connection on M associated
with the Riemannian metric ¯σ Then we can describe Newton’s law on M 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
Trang 5The Dynamics of Systems of Interacting Rigid Bodies 7-9
G
g(t)
e
Φ (1, e)
TeG v
FIGURE 7.2 The action of the exponential map on a group.
Where(1, e) is the result of the flow of a vector field 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 Te G , and it can be denoted by V The main idea underlying
this analysis is the fact that we can identify a chart inV, in the neighborhood of 0 ∈ V, with a chart in the
neighborhood of e ∈ G More specifically, 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 e x p : 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∗: Tg G −→ Tg h G , R g∗: Tg G −→ Tg h G
Next we consider the map Rg−1L 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 Rg−1L gis going to be very useful in the construction which follows:
Ad g = (Rg−1L g)e∗: Te G = V −→ Te G = V Thus, Adg is a linear map of the Lie algebraV to itself The map Ad ghas 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 e x p( f (t)) ξ = ξ + t[ f, ξ] + o(t2) (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 Rg∗and are known as pull-backs of Lg and Rg, respectively:
L∗g : T g h∗G −→ T∗
h G , R∗g : T hg∗G −→ T∗
h G
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Trang 67-10 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)
forξ ∈ T∗
g G , n ∈ Tg 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 : Te G −→ T∗
e G ), ( A ξ, n) = (An, ξ) for all ξ, n ∈ V.
Using the left translation we can define a symmetric operator Ag : Tg G −→ T∗
g G according to the relation
A g ξ = L∗
g−1AL g−1 ∗ξ Finally, via this symmetric operator we can define a metric on T g G by
ξ, ng = (Ag n, ξ) = (A g ξ, n) = n, ξ g
for allξ, n ∈ T g G The metric·, ·gis a Riemannian metric that is invariant under left translations At the
identity element e, we denote the metric on T e G = V by ·, · We can define 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 ∈ Tg (t) G The rotational velocity
with respect to the body coordinate frame is the left translation of the vector ˙g ∈ G to Te 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 = Lg−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 ∈ Tg (t) G to T e G = V, which we
denote byω s = Rg−1 ∗˙g ∈ V.
In this case we have that Ag : Tg 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 = Ag ˙g ∈ V can be expressed with respect to the body coordinate frame as M c = L∗
g M = Aωcand with respect to an inertial frame
M s = R∗
g M = Ad∗
g−1M ( Ad g∗−1is the dual of Ad g−1 ∗).
Euler’s equations according to the notation established above are given by
dω c
dt = B(ωc,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) = ωc Because the metric is left invariant, the left translation of a geodesic is also a
geodesic Thus the derivativedω c
dt depends only onω 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 Ta V, is identified
naturally withV Thus the following lemma can be stated.
Lemma 7.1 Consider the left translation L e x p(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 = T0V → V = T a V If ξ ∈ V
then
L a∗ξ = ξ + 1
2[a, ξ] + o(a2) Because geodesics can be translated to the origin using coordinates of the algebraV, the derivative d ω 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]
Trang 77-12 Robotics and Automation Handbook
With respect to an inertial coordinate frame, the velocity of the body isω s = Rg−1 ∗˙g , which when given
in matrix form yields
ω ω s 1
s 2
ω s 3
=
00 − sin(φ) sin(θ) cos(φ)cos(φ) sin(θ) sin(φ)
˙
φ
˙
θ
˙
ψ
The kinetic energy metric is given by K = 1
2 ˙g, ˙g, g = 1
2ωc, 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=
I1 0 0
0 I2 0
We can calculate the quantity Ag = L∗
g−1AL 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
I01 I0 0
2 0
− sin(θ) cos(ψ) sin(ψ) 0sin(θ) sin(ψ) cos(ψ) 0
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
¨
φI2sin(ψ)2sin(θ)2+ 2 ˙φ ˙ψ I2cos(ψ) sin(ψ) sin(θ)2− 2 ˙φ ˙ψ I1cos(ψ) sin(ψ) sin(θ)2+ ¨φI1cos(ψ)2sin(θ)2
+ 2 ˙φ ˙θ I2sin(ψ)2cos(θ) sin(θ) + 2 ˙φ ˙θ I1cos(ψ)2cos(θ) sin(θ) − 2 ˙φ ˙θ I3cos(θ) sin(θ)
− ˙ψ ˙θ I2sin(ψ)2sin(θ) + ˙ψ ˙θ I1sin(ψ)2sin(θ) + ¨θ I2cos(ψ) sin(ψ) sin(θ) − ¨θ I1cos(ψ) sin(ψ) sin(θ)
+ ˙ψ ˙θ I2cos(ψ)2sin(θ) − ˙ψ ˙θ I1cos(ψ)2sin(θ) − ˙ψ ˙θ I3sin(θ) + ¨φI3cos(θ)2
+ ˙θ2I2cos(ψ) sin(ψ) cos(θ) − ˙θ2I1cos(ψ) sin(ψ) cos(θ) + ¨ψ I3cos(θ) = T φ
− ˙φ2
I2sin(ψ)2cos(θ) sin(θ) − ˙φ2
I1cos(ψ)2cos(θ) sin(θ) + ˙φ2
I3cos(θ) sin(θ) − ˙φ ˙ψ I2sin(ψ)2sin(θ)
+ ˙φ ˙ψ I1sin(ψ)2sin(θ) + ¨φI2cos(ψ) sin(ψ) sin(θ) − ¨φI1cos(ψ) sin(ψ) sin(θ) + ˙φ ˙ψ I2cos(ψ)2sin(θ)
− ˙φ ˙ψ I1cos(ψ)2sin(θ) + ˙φ ˙ψ I3sin(θ) + ¨θ I1sin(ψ)2− 2 ˙ψ ˙θ I2cos(ψ) sin(ψ)
+ 2 ˙ψ ˙θ I1cos(ψ) sin(ψ) + ¨θ I2cos(ψ)2= Tθ
− ˙φ2
I2cos(ψ) sin(ψ) sin(θ)2+ ˙φ2
I1cos(ψ) sin(ψ) sin(θ)2+ ˙φ ˙θ I2sin(ψ)2sin(θ) − ˙φ ˙θ I1sin(ψ)2sin(θ)
− ˙φ ˙θ I2cos(ψ)2sin(θ) + ˙φ ˙θ I1cos(ψ)2sin(θ) − ˙φ ˙θ I3sin(θ) + ¨φI3cos(θ) + ˙θ2I2cos(ψ) sin(ψ)
− ˙θ2I1cos(ψ) sin(ψ) + ¨ψ I3= 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
Trang 87-14 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),
which is by definition 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
µ1: 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 is a manifold, and that there exists a map l : 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 FM and TMdenote, 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 FS, TS Thus we have
F + T = (FM + TM) ⊕ (FS + TS) We can define a priori the generalized constraint forces as those that
satisfy the condition
This is actually what B Hoffmann described in his work [20] as Kron’s method of subspaces Indeed in this work Kron’s ideas about the use of tensor mathematics in circuit analysis are applied to 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
Trang 9The Dynamics of Systems of Interacting Rigid Bodies 7-17
The degrees of freedom of the system areθ, 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 =
−m1l1g sin(θ) − F x l1cos(θ) + F y l1sin(θ)
−F x l2cos(φ) + F y l2sin(φ)
F x
−F y − m2g
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, l1sin( ˜θ) + l2sin( ˜φ), −l1cos( ˜θ) − l2cos( ˜φ)}
Nextξ∗is calculated:
ξ∗=
l1cos( ˜θ) l2cos( ˜φ)
l1sin( ˜θ) l2sin( ˜φ)
It is obvious that the mapξ is an immersion, because the rank(ξ∗ = 2 Consequently, we compute
¯
ρ = ξ∗ρ:
¯
ρ =
l2m1+ l2m2 l1l2m2cos( ˜θ − ˜φ)
l1l2m2cos( ˜θ − ˜φ) l2m2
Applyingξ∗on the 1-forms of the forces and the torques on the RHS of the dynamic equations of motion
we obtain
ξ∗(F + T) =
−m1l1g sin( ˜ θ) − l1m2g sin( ˜ θ)
−l2m2g sin( ˜ 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]
d2x i
dt2 + i
j k
d x j
dt
d x k
dt
Copyright © 2005 by CRC Press LLC
Trang 107-20 Robotics and Automation Handbook
(y1, y2)
(z1, z2)
v
–v
B u
t1
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¨z1= −vsin(θ2)
m¨z2= u − vcos(θ2)− mg
I ¨ θ = 0
m2y¨1= vsin(θ2)
m2y¨2= vcos(θ2)− m2g
I2θ¨2= u2− vτ1
Where m, I are the mass and the moment of inertia of the disk, m2, I2are the mass and the moment
of inertia of the link, and u2is 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 M1× M2 We can restrict the original system of dynamic
equations to M1× M2using the projection method developed earlier Thus we need to construct the projection functionξ : M1× M2 −→ IE(2) × IE(2) We already have a description of the projection mappings for each constrained submanifold,µ1 : M1 −→ IE(2) and ν1 : M2 −→ IE(2) We can use the combined projection function ξ = (µ1,ν1) 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(y1, y2,θ2)d y1+b(y1, y2,θ2)d y2+c(y1, y2,θ2)d θ2 However, in the equations for the 1-forms we have the variablesτ1and u2, which are different from{y1, y2,θ2} The variable u2is the input (torque) to the link, andτ1is the point of interaction between the link and the disk Thusτ1can 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
M2, thenτ1is assigned a special value because in M2,τ1is a function of{ ˜y1, ˜θ2} (the coordinates of M2)
We can compute the quantitiesξ∗andξ∗that are needed for the modeling process The projection mapξ
is defined as
ξ : {˜z1, ˜θ, ˜y1, ˜θ2} −→
˜z1, 1, ˜θ, ˜y1,1+ sin(θ2)(˜z1− ˜y1)
cos( ˜θ2) + 1, ˜θ2