Control of Robot Manipulators in Joint Space - R. Kelly, V. Santibanez and A. Loria Part 4 docx

3.17Considering these expressions, the equation of motion takes the form Equation 3.18 is the dynamic equation for robots of n DOF.. 3.2 Dynamic Model in Compact Form 73forces, which in

K(q, ˙q) = 12q˙T M (q) ˙q (3.15)

where M (q) is a matrix of dimension n ×n referred to as the inertia matrix.

M (q) is symmetric and positive definite for all q ∈ IR n The potential energy

U(q) does not have a specific form as in the case of the kinetic energy but it

is known that it depends on the vector of joint positions q.

The LagrangianL(q, ˙q), given by Equation (3.3), becomes in this case

2q˙T M (q) ˙q − ∂

∂q

1

2q˙T M (q) ˙q = M (q) ˙q (3.16)

d dt

∂

∂ ˙q

1

2q˙T M (q) ˙q = M (q)¨ q + ˙ M (q) ˙q. (3.17)Considering these expressions, the equation of motion takes the form

Equation (3.18) is the dynamic equation for robots of n DOF Notice that

(3.18) is a nonlinear vectorial differential equation of the state [q T q˙T]T

C(q, ˙q) ˙q is a vector of dimension n called the vector of centrifugal and

Coriolis forces, g(q) is a vector of dimension n of gravitational forces or

torques and τ is a vector of dimension n called the vector of external

3.2 Dynamic Model in Compact Form 73

forces, which in general corresponds to the torques and forces applied by the

actuators at the joints

The matrix C(q, ˙q) ∈ IR n ×n, called the centrifugal and Coriolis forces

matrix may be not unique, but the vector C(q, ˙q) ˙q is indeed unique One way

to obtain C(q, ˙q) is through the coefficients or so-called Christoffel symbols

of the first kind, c ijk (q), defined as

The model (3.18) may be viewed as a dynamic system with input, the

vector τ , and with outputs, the vectors q and ˙q This is illustrated in Figure

3.6

-˙q

q

Figure 3.6. Input–output representation of a robot

Each element of M (q), C(q, ˙q) and g(q) is in general, a relatively complex expression of the positions and velocities of all the joints, that is, of q and ˙q The elements of M (q), C(q, ˙q) and g(q) depend of course, on the geometry of the robot in question Notice that computation of the vector g(q) for a given

robot may be carried out with relative ease since this is given by (3.20) In

other words, the vector of gravitational torques g(q), is simply the gradient

of the potential energy functionU(q).

Example 3.5 The dynamic model of the robot from Example 3.2, that

is, Equation (3.5), may be written in the generic form (3.18) by taking

M (q) = m2l22 cos2(ϕ) ,

C(q, ˙ q) = 0 , g(q) = 0

♦

Example 3.6 The Lagrangian dynamic model of the robot

manipula-tor shown in Figure 3.4 was derived in Example 3.3 A simple tion of Equations (3.8) and (3.9) shows that the dynamic model forthis robot in compact form is

That is, the gravitational forces vector is zero ♦

The dynamic model in compact form is important because it is the modelthat we use throughout the text to design controllers and to analyze thestability, in the sense of Lyapunov, of the equilibria of the closed-loop system

In anticipation of the material in later chapters of this text and in support ofthe material of Chapter 2 it is convenient to make some remarks at this pointabout the “stability of the robot system”

In the previous examples we have seen that the model in compact formmay be rewritten in the state-space form As a matter of fact, this property isnot limited to particular examples but stands as a fact for robot manipulators

in general This is because the inertia matrix is positive definite and so is the

matrix M (q) −1; in particular, the latter always exists This is what allows us

to express the dynamic model (3.18) of any robot of n DOF in terms of the

state vector [q T q˙T]T that is, as

3.3 Dynamic Model of Robots with Friction 75

in open loop must be handled with care

We emphasize that the definition of stability in the sense of Lyapunov,which is presented in Definition 2.2 in Chapter 2, applies to an equilibrium(typically the origin) Hence, in studying the “stability of a robot manipula-tor” it is indispensable to first determine the equilibria of Equation (3.23),which describes the behavior of the robot

The necessary and sufficient condition for the existence of equilibria of

Equation (3.23), is that τ (t) be constant (say, τ ∗) and that there exist a

solution q ∗ ∈ IR n to the algebraic possibly nonlinear equation, in g(q ∗ ) = τ ∗.

In such a situation, the equilibria are given by [q T q˙T]T = [q ∗T 0T]T ∈ IR 2n.

In the particular case of τ ≡ 0, the possible equilibria of (3.23) are given

by [q T q˙T]T = [q ∗T 0T]T where q ∗ is a solution of g(q ∗) = 0 Given the

definition of g(q) as the gradient of the potential energy U(q), we see that q ∗

corresponds to the vectors where the potential energy possesses extrema

A particular case is that of robots whose workspace corresponds to the

horizontal plane In this case, g(q) = 0 and therefore it is necessary and sufficient that τ (t) = 0 for equilibria to exist Indeed, the point

q T q˙TT

=

[q ∗T 0T]T ∈ IR 2n where q ∗ is any vector of dimension n is an equilibrium.

This means that there exist an infinite number of equilibria See also Example2.2

The development above makes it clear that if one wants to study the topic

of robot stability in open loop (that is, without control) one must specify thedynamic model as well as the conditions for equilibria to exist and only then,select one among these equilibria, whose stability is of interest Consequently,the question “is the robot stable? ” is ambiguous in the present context

3.3 Dynamic Model of Robots with Friction

It is important to notice that the generic Equation (3.18) supposes that thelinks are rigid, that is, they do not present any torsion or any other defor-mation phenomena On the other hand, we also considered that the jointsbetween each pair of links are stiff and frictionless The incorporation of thesephenomena in the dynamic model of robots is presented in this and the fol-lowing section

Friction effects in mechanical systems are phenomena that depend on tiple factors such as the nature of the materials in contact, lubrication of the

mul-latter, temperature, etc For this reason, typically only approximate models of

friction forces and torques are available Yet, it is accepted that these forcesand torques depend on the relative velocity between the bodies in contact.Thus, we distinguish two families of friction models: the static models, in

which the friction force or torque depends on the instantaneous relative

ve-locity between bodies and, dynamic models, which depend on the past values

of the relative velocity

Thus, in the static models, friction is modeled by a vector f ( ˙q) ∈ IR nthat

depends only on the joint velocity ˙q Friction effects are local, that is, f ( ˙q)

A “classical” static friction model is one that combines the so-called viscous

and Coulomb friction phenomena This model establishes that the vector f ( ˙q)

is given by

f( ˙q) = F m1q + F˙ m2 sign( ˙q) (3.24)

where F m1and F m2are n ×n diagonal positive definite matrices The elements

of the diagonal of F m1correspond to the viscous friction parameters while the

elements of F m2correspond to the Coulomb friction parameters Furthermore,

in the model given by (3.24)

However, sign(0) is undefined in the sense that one do not associate a

partic-ular real number to the “function” sign(x) when x = 0.

In certain applications, this fact is not of much practical relevance, as forinstance, in velocity regulation, – when it is desired to maintain an operatingpoint involving high and medium-high constant velocity, but the definition of

3.4 Dynamic Model of Elastic-joint Robots 77

sign(0) is crucial both from theoretical and practical viewpoints, in position

control (i.e when the control objective is to maintain a constant position).

For this reason, and in view of the fact that the “classical” model (3.24)describes inadequately the behavior of friction at very low velocities, that is,when bodies are at rest and start to move, this is not recommended to modelfriction when dealing with the position control problem (regulation) In thiscase it is advisable to use available dynamic models The study of such models

is beyond the scope of this textbook

Considering friction in the joints, the general dynamic equation of themanipulator is now given by

M (q)¨ q + C(q, ˙q) ˙q + g(q) + f( ˙q) = τ (3.25)

In general, in this text we shall not assume that friction is present in thedynamic model unless it is explicitly mentioned In such a case, we consideronly viscous friction

3.4 Dynamic Model of Elastic-joint Robots

In many industrial robots, the motion produced by the actuators is ted to the links via gears and belts These, are not completely stiff but theyhave elasticity which can be compared to that of a spring In the case of revo-lute joints, where the actuators are generally electric motors these phenomenaboil down to a torsion in the axis that connects the link to the rotor of themotor The elasticity effect in the joints is more noticeable in robots whichundergo displacements with abrupt changes in velocity A direct consequence

transmit-of this effect, is the degradation transmit-of precision in the desired motion transmit-of the robot.Evidently, industrial robots are designed in a way to favor the reduction ofjoint elasticity, however, as mentioned above, such an effect is always present

to some degree on practically any mechanical device An exception to this

rule is the case of the so-called direct-drive robots, in which the actuators are

directly connected to the links

Robot dynamics and control under the consideration of joint elasticity, hasbeen an important topic of research since the mid-1980s and continues today

We present below only a brief discussion

Consider a robot manipulator composed of rigid n links connected through

revolute joints Assume that the motion of each link is furnished by electric

motors and transmitted via a set of gears Denote by J i, the moment of inertia

of the rotors about their respective rotating axes Let r i be the gear reduction

ratio of each rotor; e.g if r = 50 we say that for every 50 rotor revolutions, the

axis after the corresponding gear undergoes only one full turn Joint elasticitybetween each link and the corresponding axis after the set of gears is modeled

via a torsional spring of constant torsional ‘stiffness’, k The larger k, the

stiffer the joint Figure 3.7 illustrates the case of a robot with two joints.

The joint positions of each link are denoted, as usual by q while the angular positions of the axes after the set of gears are θ = [θ1 θ2 · · · θ n]T Due to

the elasticity, and while the robot is in motion we have, in general, q

Figure 3.7. Diagram of a robot with elastic joints

Typically, the position and velocity sensors are located at the level of the

rotors’ axes Thus knowing the gears reduction rate, only θ may be determined and in particular, q is not available This observation is of special relevance in control design since the variable to be controlled is precisely q, which cannot

be measured directly unless one is able to collocate appropriate sensors tomeasure the links’ positions, giving a higher manufacturing cost

Due to elasticity a given robot having n links has 2n DOF Its generalized

coordinates are [q T θ T]T The kinetic energy function of a robot with elasticjoints corresponds basically to the sum of the kinetic energies of the links and

3.4 Dynamic Model of Elastic-joint Robots 79

those of the rotors,4that is,

K(q, ˙q, ˙θ) = 1

2q˙T M (q) ˙q +1

2˙θ T J ˙θ where M (q) is the inertia matrix of the “rigid” (that is, assuming an infinite

stiffness value of k i for all i) robot, and J is a diagonal positive definite

matrix, whose elements are the moments of inertia of the rotors, multiplied

by the square of the gear reduction ratio, i.e.

to that of the robot as if the joints were absolutely stiff The matrix K is diagonal positive definite and its elements are the ‘torsion constants’, i.e.

Finally, using (3.16), (3.17), (3.19) and

through the state vector [q T θ T q˙T ˙θ T]T as

Example 3.7 Consider the device shown in Figure 3.8, which consists

of one rigid link of mass m, and whose center of mass is localized

at a distance l from the rotation axis The moment of inertia of the

link with respect to the axis that passes through its center of mass is

denoted by I The joint is elastic and has a torsional constant k The rotor’s inertia is denoted by J

The dynamic model of this device may be computed noticing that

3.4 Dynamic Model of Elastic-joint Robots 81

Figure 3.8.Link with an elastic joint

Unless clearly stated otherwise, in this text we consider only robots with

stiff joints i.e the model that we use throughout this textbook is given by

q1

v2

q2

Figure 3.10. Example of a 2-DOF robot

3.5 Dynamic Model of Robots with Actuators

On a real robot manipulator the torques vector τ , is delivered by actuators

that are typically electromechanical, pneumatic or hydraulic Such actuatorshave their own dynamics, that is, the torque or force delivered is the product

of a dynamic ‘transformation’ of the input to the actuator This input may

be a voltage or a current in the case of electromechanical actuators, fluid(typically oil) flux or pressure in the case of hydraulic actuators In Figures3.9 and 3.10 we illustrate two robotic arms with 2 DOF which have actuatorstransmitting the motion through gears in the first case, and through gear andbelt in the second

Actuators with Linear Dynamics

In certain situations, some types of electromechanical actuators may be eled via second-order linear differential equations

mod-A common case is that of direct-current (DC) motors The dynamic model

which relates the input voltage v applied to the motor’s armature, to the put torque τ delivered by the motor, is presented in some detail in Appendix

out-3.5 Dynamic Model of Robots with Actuators 83

D A simplified linear dynamic model of a DC motor with negligible armatureinductance, as shown in Figure 3.11, is given by Equation (D.16) in AppendixD,

• K b : back emf [V s/rad],

• f m: rotor friction coefficient with respect to its hinges [N m],

• τ : net applied torque after the set of gears at the load axis [N m],

• q : angular position of the load axis [rad],

• r : gear reduction ratio (in general r  1),

• v : armature voltage [V].

Equation (3.29) relates the voltage v applied to the armature of the motor

to the torque τ applied to the load, in terms of its angular position, velocity

Figure 3.11.Diagram of a DC motor

Considering that each of the n joints is driven by a DC motor we obtain

R = diag

1

where for each motor (i = 1, · · · , n), J m i corresponds to the rotor inertia, f m i

to the damping coefficient, (K a K b /R a)ito an electromechanical constant and

r i to the gear reduction ratio

Thus, the complete dynamic model of a manipulator (considering friction

in the joints) and having its actuators located at the joints6 is obtained by

substituting τ from (3.30) in (3.25),

(R M (q) + J ) ¨ q + R C(q, ˙q) ˙q + R g(q) + R f( ˙q) + B ˙q = Kv (3.32)The equation above may be considered as a dynamic system whose input

is v and whose outputs are q and ˙q A block-diagram for the model of the

manipulator with actuators, given by (3.32), is depicted in Figure 3.12

˙

q

q τ

v

¨

q

Figure 3.12. Block-diagram of a robot with its actuators

Example 3.8 Consider the pendulum depicted in Figure 3.13 The

de-vice consists of a DC motor coupled mechanically through a set ofgears, to a pendular arm moving on a vertical plane under the action

• m b : arm mass (without load);

6 Again, we neglect the gyroscopic and other coupling effects between the rotorsand the links

3.5 Dynamic Model of Robots with Actuators 85

m b , J

m q

Figure 3.13.Pendular device with a DC motor

• l b : distance from the rotating axis to the arm center of mass(without load);

• m : load mass at the tip of the arm (assumed to be punctual7);

• l : distance from the rotating axis to the load m;

• g : gravity acceleration;

• τ : applied torque at the rotating axis;

• f L : friction coefficient of the arm and its load

The equation above may also be written in the compact form

J L q + f¨ L q + k˙ L sin(q) = τ

where

J L = J + ml2

k L = [m b l b + ml]g

Hence, the complete dynamic model of the pendular device may

be obtained by substituting τ from the model of the DC motor, (3.29),

in the equation of the pendular arm, i.e.

from which, by simple inspection, comparing to (3.32), we identify

7That is, it is all concentrated at a point – the center of mass – and has no “size”

or shape

The robot-with-actuators Equation (3.32) may be simplified considerably

when the gear ratios r i are sufficiently large In such case (r i  1), we have

R ≈ 0 and Equation (3.32) may be approximated by

J ¨ q + B ˙q ≈ Kv

That is, the nonlinear dynamics (3.25) of the robot may be neglected This can

be explained in the following way If the gear reduction ratio is large enough,then the associated dynamics of the robot-with-actuators is described only

by the dynamics of the actuators This is the main argument that supportsthe idea that a good controller for the actuators is also appropriate to controlrobots having such actuators and geared transmissions with a high reductionratio

It is important to remark that the parameters involved in Equation (3.30)depend exclusively on the actuators, and not on the manipulator nor on its

load Therefore, it is reasonable to assume that such parameters are constant

The existence of the matrix M  (q) −1allows us to express the model (3.33)

in terms of the state vector [q ˙q] as

That is, the input torque becomes simply a voltage scaled by R −1 K At

this point and in view of the scope of this textbook we may already formulatethe problem of motion control for the system above, under the conditions

In anticipation of the material in later chapters of this text and in. .. C(q, ˙q) and g(q) is in general, a relatively complex expression of the positions and velocities of all the joints, that is, of q and ˙q The elements of M (q), C(q, ˙q) and g(q) depend of course,...

In certain applications, this fact is not of much practical relevance, as forinstance, in velocity regulation, – when it is desired to maintain an operatingpoint involving high and medium-high