Kurfess T.R. Robotics and Automation Handbook

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-8 Robotics and Automation Handbook

The Lagrangian for the manipulator is the difference between the kinetic and potential energies, that is,

L(q,

q) =

M(q)

q − V(q) =



i, j =1

(q)

− V(q)

where M

(q) is the component (i, j) of the manipulator inertia matrix at q. Substituting these expressions

into the Euler-Lagrange Equations (5.10) we obtain



j=1

(q)



j,k=1



ijk

(q)

∂V

∂q

(q) = Y

, i ∈{1, ..., n}

where the functions 

ijk

are the Christoffel symbols (of the ﬁrst kind) of the inertia matrix M and are deﬁned



ijk

(q) =



∂ M

(q)

∂q

∂ M

(q)

∂q

−

∂ M

(q)

∂q



(5.13)

Collecting the equations in a vector format, we obtain

M(q)

q + C(q,

q + G(q) = Y (5.14)

where C(q,

q) is the Coriolis matrix for the manipulator with components

(q,

q) =



k=1



ijk

(q)

and where G

(q) =

∂V

∂q

(q). Equation (5.14) suggests that the robot dynamics consists of four components:

the inertial forces M(q)

q, the Coriolis and centrifugal forces C(q,

q, the conservative forces G(q), and

the generalized force Y composed of all the non-conservative external forces acting on the manipulator.

The Coriolis and centrifugal forces depend quadratically on the generalized velocities

and reﬂect the

inter-link dynamic interactions. Traditionally, terms involving products

, i = j are called Coriolis

forces, while centrifugal forces have terms of the form (

)

A direct calculation can be used to show that the robot dynamics has the following two properties [29].

For all q ∈ Q

1. the manipulator inertia matrix M(q) is symmetric and positive-deﬁnite, and

2. the matrix (

M(q) − 2C(q,

q)) ∈ R

n×n

is skew-symmetric.

The ﬁrst property is a mathematical statement of the following fact: the kinetic energy of a system is a

quadratic form which is positive unless the system is at rest. The second property is referred to as the

passivity property of rigid linkages; this property implies that the total energy of the system is conserved

in the absence of friction. The property plays an important role in stability analysis of many robot control

schemes.

5.3.3 Generalized Force Computation

Givenaconﬁguration manifold Q, the tangent bundle is the set TQ =

{

(q,

q)| q ∈ Q

}

. The Lagrangian

is formally a real-valued map on the tangent bundle, L : TQ → R. Recall that given a scalar function on

a vector space, its partial derivative is a map from the dual space to the reals. Similarly, it is possible to

interpret the partial derivatives

∂ L

∂q

and

∂ L

∂

as functions taking values in the dual of the tangent bundle,

∗

Q. Accordingly, the Euler-Lagrange equations in vector form

∂ L

∂

−

∂ L

∂q

= Y

Lagrangian Dynamics 5

-9

can be interpreted as an equality on T

∗

Q. The external force Y is thus formally a one-form, i.e., a map

Y : Q →T

∗

Q.WeletY

denote the ith component of Y . Roughly speaking, Y

is the component of

generalized force Y that directly affects the coordinate q

. In what follows we derive an expression for the

generalized force Y

as a function of the wrenches acting on the individual links.

Let us start by introducing two useful concepts. Recall that given two manifolds Q

and Q

and a smooth

map φ : Q

→ Q

, the tangent map Tφ : TQ

→TQ

is a linear function that maps tangent vectors from

to T

φ(q)

.IfX ∈ T

is a tangent vector and γ : R → Q

is a smooth curve tangent to X at q,

then Tφ(X) is the vector tangent to the curve φ ◦γ at φ(q). In coordinates this linear map is the Jacobian

matrix of φ. Given a one-form ω on Q

, the pull-back of ω is the one-form (φ

∗

ω)onQ

deﬁned by

(φ

∗

ω)(q); X=ω; T

φ(X)

for all q ∈ Q

and X ∈ T

Now consider a linkage consisting of n rigid bodies with the conﬁguration space Q. Recall that the

extended forward kinematics function κ : Q → SE

(3) from Equation (5.5) maps a conﬁguration q ∈ Q

to the conﬁguration of each of the links, i.e., to a vector (g

(q), ..., g

(q)) in the Cartesian space

(3). Forces and velocities in the Cartesian space are described as twists and wrenches; they are thus

elements of se

(3) and se

∗

(3), respectively. Let W

be the wrench acting on the ith link. The total force

in the Cartesian space is thus W = (W

, ..., W

). The generalized force Y

is the component of the total

conﬁguration space force Y in the direction of q

. Formally, Y

=Y ;

∂

∂q

,where

∂

∂q

is the ith coordinate

vector ﬁeld.

It turns out that the total conﬁguration space force Y is the pull-back of the total Cartesian space force

W to T

∗

Q through the extended forward kinematics map, i.e.,

Y = κ

∗

(W)

Using this relation and the deﬁnition of the pull-back we thus have

(q) =



Y(q);

∂

∂q





W(κ(q)); T



∂

∂q



(5.15)

It is well established that in the absence of external forces, the generalized force Y

is the torque applied

at the ith joint if the joint is revolute and the force applied at the ith joint if the joint is prismatic; let us

formally derive this result using Equation (5.15). Consider a serial linkage with the forward kinematics

map given by (5.7). Assume that the ith joint connecting (i − 1)th and ith link is revolute and let τ

the torque applied at the joint. At q ∈ Q,wecompute



∂

∂q



= (0, ...,0

  

j−1

, ξ

S, j

(q), ..., ξ

S, j

(q))

where ξ

S, j

(q)isdeﬁned by Equation (5.6) and it is the unit twist corresponding to a rotation about the j th

joint axis. The wrench in the Cartesian space resulting from the torque τ

acting along the ith joint axis is

W = τ

(0, ...,0

  

i−2

, −σ

S,i

(q), σ

S,i

(q), 0, ..., 0), where σ

S,i

(q) is the unit wrench corresponding to a torque

about the ith joint axis. It is then easy to check that W results in



0, j = i,

, j = i

-10 Robotics and Automation Handbook

5.3.4 Geometric Interpretation

Additional differential geometric insight into the dynamics can be gained by observing that the kinetic

energy provides a Riemannian metric on the conﬁguration manifold Q, i.e., a smoothly changing rule for

computing the inner product between tangent vectors. Recall that given (q,

q) ∈ TQ, the kinetic energy of

a manipulator is T(q,

q) =

M(q)

q,whereM(q) is a positive-deﬁnite matrix. If we have two vectors

, v

∈ T

Q, we can thus deﬁne their inner product by v

, v

 =

)

M(q)v

. Physical properties of

a manipulator guarantee that M(q) and thus the rule for computing the inner product changes smoothly

over Q. For additional material on this section we refer to [11].

Given two smooth vector ﬁelds X and Y on Q, the covariant derivative of Y with respect to X is the

vector ﬁeld ∇

Y with coordinates

(∇

∂Y

∂q





where X

and Y

are the ith and j th component of X and Y , respectively. The operator ∇ is called an afﬁne

connection and it is determined by the n

functions





. A direct calculation shows that for real valued

functions f and g deﬁned over Q, and vector ﬁelds X, Y, and Z:

∇

fX+gY

Z = f ∇

Z + g ∇

∇

(Y + Z) =∇

Y +∇

∇

( fY) = f ∇

Y + X( f )Y

where in coordinates, X( f ) =



i=1

∂ f

∂q

When the functions





are computed according to





(q) =



l=1



∂ M

(q)

∂q

∂ M

(q)

∂q

−

∂ M

(q)

∂q



(5.16)

where M

(q) arethe componentsof M

−1

(q), they arecalled the Christoffelsymbols(of the secondkind) of the

Riemannian metric M, and the afﬁne connection is called the Levi-Civita connection. From Equation (5.13)

and Equation (5.16), it is easy to see that for a Levi-Civita connection,



ijk



l=1





Assume that the potential forces are not present so that L(q,

q) = T(q,

q). The manipulator dynamics

equations (5.14) can be thus written as:

q + M

−1

(q)C(q,

q = M

−1

(q)Y

It can be seen that the last equation can be written in coordinates as





(q)

= F

and in vector format as

∇

q = F (5.17)

where∇istheLevi-Civita connectioncorrespondingto M(q)and F = M

−1

(q)Y isthe vectorﬁeldobtained

from the one-form Y through the identity Y ; X=F , X. In the absence of the external forces, Equation

(5.17) becomes the so-called geodesic equation for the metric M. It is a second order differential equation

whose solutions are curves of locally minimal length – like straight lines in R

or great circles on a sphere.

It can also be shown that among all the curves γ : [0, 1] → Q that connect two given points, a geodesic is

Lagrangian Dynamics 5

-11

the curve that minimizes the energy integral:

E (γ ) =



T(γ (t),

γ (t))dt

This geometric insight implies that if a mechanical system moves freely with no external forces present, it

will move along the minimum energy paths.

When the potential energy is present, Equation (5.17) still applies if the resulting conservative forces

are included in F . The one-form Y describing the conservative forces associated with the potential energy

V is the differential of V, that is, Y =−dV, where the differential dV is deﬁned by dV; X=X(V).

The corresponding vector ﬁeld F =−M

−1

(q)dV is related to the notion of gradient of V,speciﬁcally,

F =−grad V . In coordinates, differential and gradient of V have components:

(dV)

∂V

∂q

, and (grad V)

= M

∂V

∂q

5.4 Constrained Systems

One of the advantages of Lagrange’s formalism is a systematic procedure to deal with constraints. General-

ized coordinates in themselves are a way of dealing with constraints. For example, for robot manipulators,

joints limit the relative motion between the links and thus represent constraints. We can avoid modeling

these constraints explicitly by choosing joint variables as generalized coordinates. However, in robotics it

is often necessary to constrain the motion of the end-effector in some way, model rolling of the wheels

of a mobile robot without slippage or, for example, taking into account various conserved quantities for

space robots. Constraints of this nature are external to the robot itself so it is desirable to model them

explicitly. Mathematically, a constraint can be described by an equation of the form ϕ(q,

q) = 0, where

q ∈R

are the generalized coordinates and ϕ : TQ →R

. However, there is a fundamental difference

between constraints of the form ϕ(q) = 0, called holonomic constraints, and those where the dependence

on the generalized velocities cannot be eliminated through the integration, known as nonholonomic con-

straints. Also, among the nonholonomic constraints, only those that are linear in the generalized velocities

turn out to be interesting in practice. In other words, typical nonholonomic constraints are of the form

ϕ(q,

q) = A(q)

Holonomic constraints restrict the motion of the system to a submanifold of the conﬁguration manifold

Q. For nonholonomic constraints such a constraint manifold does not exist. However, formally the

holonomic and nonholonomic constraints can be treated in the same way by observing that a holonomic

constraint ϕ(q) = 0 can be differentiated to obtain A(q)

q = 0, where A(q) =

∂ϕ

∂q

∈ R

m×n

. The general

form of constraints can be thus assumed to be

A(q)

q = 0 (5.18)

We will assume that A(q) has full rank everywhere.

Constraints are enforced on the mechanical system via a constraint force. A typical assumption is that

the constraint force does no work on the system. This excludes examples such as the end-effector sliding

with friction along a constraint surface. With this assumption, the constraint force must be at all times

perpendicular to the velocity of the system (strictly speaking, it must be in the annihilator of the set of

admissible velocities). Since the velocity satisﬁes Equation (5.18), the constraint force, say ,willbeofthe

form

 = A

(q)λ (5.19)

-12 Robotics and Automation Handbook

where λ ∈R

is a set of Lagrange multipliers. The constrained Euler-Lagrange equations thus take the

form:

∂ L

∂

−

∂ L

∂q

= Y + A

(q)λ (5.20)

These equations, together with the constraints (5.18), determine all the unknown quantities (either Y and

λ, if the motion of the system is given, or trajectories for q and λ,ifY is given).

Several methods can be used to eliminate the Lagrange multipliers and simplify the set of equa-

tions (5.18)–(5.20). For example, let S(q) be a matrix whose columns are the basis of the null-space

of A(q). Thus we can write

q = S(q)η (5.21)

for some vector η ∈ R

n−m

. The components of η are called pseudo-velocities. Now multiply (5.20) on the

left with S

(q) and use Equation (5.21) to eliminate

q. The dynamic equations then become

(q)M(q)S(q)

η + S

(q)M(q)

∂ S(q)

∂q

S(q)η + S

(q)C(q, S(q)η) + S

(q)G(q) = S

(q)Y (5.22)

The last equation together withEquation (5.21) then completely describesthe system subject to constraints.

Sometimes, such procedures are also known as embedding of constraints into the dynamic equations.

5.4.1 Geometric Interpretation

The constrained Euler-Lagrange equation can also be given an interesting geometric interpretation. As

we did in Section 5.3.4, consider a manifold Q with the Riemannian metric M. Equation (5.18) allows

the system to move only in the directions given by the null-space of A(q). Geometrically, the velocity of

the system at each point q ∈ Q must lie in the subset D(q) =∅A(q) of the tangent space T

Q. Formally,

D =∪

q∈Q

D(q)isadistribution on Q. For obvious reasons, this distribution will be called the constraint

distribution. Note that given a constraint distribution D, Equation (5.18) simply becomes

q ∈D(q).

Let P : TQ → Ddenote the orthogonal (with respect to the metric M) projection onto the distribution

of feasible velocities D. In other words, at each q ∈ Q, P (q) maps T

Q into D(q) ⊂ T

Q.LetD

⊥

denote

the orthogonal complement to D with respect to the metric M and let P

⊥

= I − P ,whereI is the identity

map on TQ. Geometrically, the Equation (5.18) and Equation (5.20) can be written as:

∇

q = (t) + F (5.23)

⊥

(

q) = 0 (5.24)

where (t) ∈D

⊥

is the same as that in Equation (5.19) and is the Lagrange multiplier enforcing the

constraint.

It is known, e.g., see [9, 17], that Equation (5.23) can be written as:



∇

q = P (Y) (5.25)

where



∇ is the afﬁne connection given by



∇

Y =∇

Y +∇

⊥

(Y)) − P

⊥

(∇

Y) (5.26)

We refer to the afﬁne connection



∇ as the constraint connection. Typically, the connection



∇ is only applied

to the vector ﬁelds that belong to D. A short computation shows that for Y ∈ D,



∇

Y = P (∇

Y) (5.27)

Lagrangian Dynamics 5

-13

The last expression allows us to evaluate



∇

Y directly and thus avoid signiﬁcant amounts of computation

needed to explicitly compute



∇ from Equation (5.26). Also, by choosing a basis for D, we can directly

derive Equation (5.22) from Equation (5.25).

The important implication of this result is that even when a system is subject to holonomic or nonholo-

nomic constraints, it is indeed possible to write the equations of motion in the form (5.17). In every case

the systems evolve over the same manifold Q, but they are described with different afﬁne connections.

This observation has important implications for control of mechanical systems in general and methods

developed for the systems in the form (5.17) can be quite often directly applied to systems with constraints.

5.5 Impact Equations

Quite often, robot systems undergo impacts as they move. A typical example is walking robots, but impacts

also commonly occur during manipulation and assembly. In order to analyze and effectively control such

systems, it is necessary to have a systematic procedure for deriving the impact equations. We describe a

model for elastic and plastic impacts.

In order to derive the impact equations, recall that the Euler-Lagrange Equations (5.10) are derived

from the Lagrange-d’Alembert principle. First, for a given C

curve γ :[a, b] → Q,deﬁne a variation

σ :(−, ) × [a, b] → Q,aC

map with the properties:

1. σ (0, t) = γ (t);

2. σ (s, a) = γ (a), σ (s, b) = γ (b)

Let δq =

s=0

σ (s, t). Note that δq is a vectorﬁeld along γ . A curve γ (t) satisﬁes the Lagrange-d’Alembert

Principle for the force Y and the Lagrangian L (q,

q) if for every variation σ (s , t):



s=0





σ (s, t),

σ (s, t)



dt +



Y; δqdt = 0 (5.28)

Now assume a robot moving on a submanifold M

⊆ Q before the impact, on M

⊆ Q after the impact,

with the impact occurring on a M

⊆ Q,whereM

⊆ M

and M

⊆ M

. Typically, we would have M

= M

when the system bounces off M

(impact with some degree of restitution) or M

= M

⊆ M

for a plastic

impact. Assume that M

=ϕ

−1

(0), where ϕ

: Q →R

. In other words, we assume that 0 is a regular value

of ϕ

and that the submanifold M

corresponds to the zero set of ϕ

. The value p

is the co-dimension of

the submanifold M

.Letτ be the time at which the impact occurs. A direct application of (5.28) leads to

the following equation:





∂ L

∂q

−

∂ L

∂



δqdt+



Y; δq dt +



∂ L

∂



t=τ

−

∂ L

∂



t=τ



δq

t=τ

= 0

Given that δq is arbitrary, the ﬁrst two terms would yield the constrained Euler-Lagrange Equations (5.20).

The impact equations can be derived from the last term:



∂ L

∂



t=τ

−

∂ L

∂



t=τ



δq

t=τ

= 0 (5.29)

Since γ (s, τ ) ∈ M

,wehavedϕ

· δq

t=τ

= 0. Equation (5.29) thus implies



∂ L

∂



t=τ

−

∂ L

∂



t=τ



∈ span dϕ

(5.30)

-14 Robotics and Automation Handbook

The expression

∂ L

∂

can be recognized as the momentum. Equation (5.30) thus leads to the expected

geometric interpretation: the momentum of the system during the impact can change only in the directions

“perpendicular” to the surface on which the impact occurs.

Let ϕ

, ..., ϕ

bethecomponentsofϕ

,andlet grad ϕ

bethegradientof ϕ

.Observethat 

∂ L(q,

∂

; V=



q , V. Equation (5.30) can be therefore written as:

(

t=τ

−

t=τ

) =



j=1

3 j

grad ϕ

(5.31)

In the case of a bounce, the equation suggests that the velocity in the direction orthogonal (with respect

to the Riemannian metric M) to the impact surface undergoes the change; the amount of change depends

on the coefﬁcient of restitution. In the case of plastic impact, the velocity should be orthogonally (again,

with respect to the metric M) projected to the impact surface. In other words, using appropriate geometric

tools, the impact can indeed be given the interpretation that agrees with the simple setting typically taught

in the introductory physics courses.

5.6 Bibliographic Remarks

The literature on robot dynamics is vast and the following list is just a small sample. A good starting point

to learn about robot kinematics and dynamics is many excellent robotics textbooks [3, 10, 21, 26, 29]. The

earliest formulations of the Lagrange equations of motions for robot manipulators are usually considered

to be [32] and [15]. Computationally efﬁcient procedures for numerically formulating dynamic equations

of serial manipulators using Newton-Euler formalism are described in [19, 22, 30], and in the Lagrange’s

formalism in [14]. A comparison between the two approaches is given in [28]. These algorithms have been

generalized to more complex structures in [4, 12, 18]. A good review of robot dynamics algorithms is [13].

Differential geometric aspects of modeling and control of mechanical systems are the subjects of [1, 6,

9, 20, 27].

Classical dynamics algorithms are recast using differential geometric tools in [31] and [24] for serial

linkages and in [25] for linkages containing closed kinematic chains.

Control and passivity of robot manipulators and electromechanical systems are discussed in [2, 23].

Impacts are extensively studied in [8]. An interesting analysis is also presented in [16].

5.7 Conclusion

The chapter gives an overview of the Lagrange’s formalism for deriving the dynamic equations for robotic

systems. The emphasis is on the geometric interpretation of the classical results as such interpretation leads

to a very intuitive and compact treatment of constraints and impacts. We start by providing a brief overview

of kinematics of serial chains. Rigid-body transformations are described with exponential coordinates;

the product of exponentials formula is the basis for deriving forward kinematics map and for the velocity

analysis. The chosen formalism allows us to provide a direct mathematical interpretation to many classical

notions from screw calculus. Euler-Lagrange equations for serial linkages are presented next. First, they

are stated in the traditional coordinate form. The equations are then rewritten in the Riemannian setting

using the concept of afﬁne connections. This form of Euler-Lagrange equations enables us to highlight

their variational nature. Next, we study systems with constraints. Again, the constrained Euler-Lagrange

equations are ﬁrst presented in their coordinate form. We then show how they can be rewritten using

the concept of a constraint connection. The remarkable outcome of this procedure is that the equations

describing systems with constraints have exactly the same form as those for unconstrained systems; what

changes is the afﬁne connection that is used. We conclude the chapter with a derivation of equations for

systems that undergo impacts. Using geometric tools we show that the impact equations have a natural

interpretation as a linear velocity projection on the tangent space of the impact manifold.

Lagrangian Dynamics 5

-15

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Kane’s Method in

Robotics

Keith W. Bufﬁnton

Bucknell University

6.1 Introduction

6.2 The Essence of Kane’s Method

6.3 Two DOF Planar Robot with Two Revolute Joints

Preliminaries

•

Generalized Coordinates and Speeds

•

Velocities

•

Partial Velocities

•

Accelerations

•

Generalized

Inertia Forces

•

Generalized Active Forces

•

Equations of

Motion

•

Additional Considerations

6.4 Two-DOF Planar Robot with One Revolute Joint

and One Prismatic Joint

Preliminaries

•

Generalized Coordinates and Speeds

•

Velocities

•

Partial Velocities

•

Accelerations

•

Generalized

Inertia Forces

•

Generalized Active Forces

•

Equations

of Motion

6.5 Special Issues in Kane’s Method

Linearized Equations

•

Systems Subject to Constraints

•

Continuous Systems

6.6 Kane’s Equations in the Robotics Literature

6.7 Commercially Available Software Packages Related

to Kane’s Method

SD/FAST

•

Pro/ENGINEER Simulation (formerly

Pro/MECHANICA)

•

AUTOLEV

•

ADAMS

•

Working

Model

6.8 Summary

6.1 Introduction

In 1961, Professor Thomas Kane of Stanford University published a paper entitled “Dynamics of Nonholo-

nomic Systems” [1] that described a method for formulating equations of motion for complex dynamical

systems that was equally applicable to either holonomic or nonholonomic systems. As opposed to the

use of Lagrange’s equations, this new method allowed dynamical equations to be generated without the

differentiation of kinetic and potential energy functions and, for nonholonomic systems, without the need

to introduce Lagrange multipliers. This method was later complemented by the publication in 1965 of a

paper by Kane and Wang entitled “On the Derivation of Equations of Motion” [2] that described the use

of motion variables (later called generalized speeds) that could be any combinations of the time deriva-

tives of the generalized coordinates that describe the conﬁguration of a system. These two papers laid the

foundation for what has since become known as Kane’s method for formulating dynamical equations.