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4.8 THE ANALYTICAL JACOBIAN

The Jacobian matrix derived above is sometimes called the Geometric Jacobian to distinguish it from the Analytical Jacobian, denoted Ja(q), considered in this section, which is based on a minimal representation for the orientation of the end-e↵ector frame. Let

X =

 d(q)

↵(q) (4.99)

denote the end-e↵ector pose, where d(q) is the usual vector from the origin of the base frame to the origin of the end-e↵ector frame and ↵ denotes a minimal representation for the orientation of the end-e↵ector frame relative to the base frame. For example, let ↵ = [ , ✓, ]^T be a vector of Euler angles as defined in Chapter 2. Then we look for an expression of the form

X =˙  ˙d

˙↵ = Ja(q) ˙q (4.100)

to define the analytical Jacobian.

It can be shown (Problem 4-9) that, if R = Rz, Ry,✓Rz, is the Euler angle transformation then

˙R = S(!)R (4.101)

in which !, defining the angular velocity is given by

! =

The components of ! are called the nutation, spin, and precession, respectively.

Combining the above relationship with the previous definition of the Jacobian,

i.e. 

Jacobian as

Ja(q) =

I 0

0 B(↵) ¹ J(q) (4.108)

provided det B(↵) 6= 0.

In the next section we discuss the notion of Jacobian singularities, which are configurations where the Jacobian loses rank. Singularities of the matrix B(↵) are called representational singularities. It can easily be shown (Problem 4-20) that B(↵) is invertible provided s✓ 6= 0. This means that the singularities of the analytical Jacobian include the singularities of the geometric Jacobian, J, as defined in the next section, together with the representational singularities.

4.9 SINGULARITIES

The 6 ⇥ n Jacobian J(q) defines a mapping

⇠ = J(q)˙q (4.109)

between the vector ˙q of joint velocities and the vector ⇠ = (v, !)^T of end-e↵ector velocities. This implies that the all possible end-end-e↵ector velocities are linear combinations of the columns of the Jacobian matrix,

⇠ = J1˙q1+ J2˙q2· · · + Jⁿ˙qn

For example, for the two-link planar arm, the Jacobian matrix given in Equation (4.86) has two columns. It is easy to see that the linear velocity of the end-e↵ector must lie in the xy-plane, since neither column has a nonzero entry for the third row. Since ⇠ 2 R⁶, it is necessary that J have six linearly independent columns for the end-e↵ector to be able to achieve any arbitrary velocity (see Appendix B).

The rank of a matrix is the number of linearly independent columns (or rows) in the matrix. Thus, when rank J = 6, the end-e↵ector can execute any arbitrary velocity. For a matrix J 2 R^6⇥n, it is always the case that rank J  min(6, n). For example, for the two-link planar arm, we always have rank J  2, while for an anthropomorphic arm with spherical wrist we always have rank J  6.

The rank of a matrix is not necessarily constant. Indeed, the rank of the manipulator Jacobian matrix will depend on the configuration q. Configurations for which the rank J(q) is less than its maximum value are called singularities or singular configurations. Identifying manipulator singularities is important for several reasons.

1. Singularities represent configurations from which certain directions of motion may be unattainable.

2. At singularities, bounded end-e↵ector velocities may correspond to unbounded joint velocities.

SINGULARITIES 133 3. At singularities, bounded end-e↵ector forces and torques may correspond to

unbounded joint torques. (We will see this in Chapter 9).

4. Singularities usually (but not always) correspond to points on the boundary of the manipulator workspace, that is, to points of maximum reach of the manipulator.

5. Singularities correspond to points in the manipulator workspace that may be unreachable under small perturbations of the link parameters, such as length, o↵set, etc.

6. Near singularities there will not exist a unique solution to the inverse kine-matics problem. In such cases there may be no solution or there may be infinitely many solutions.

There are a number of methods that can be used to determine the singu-larities of the Jacobian. In this chapter, we will exploit the fact that a square matrix is singular when its determinant is equal to zero. In general, it is difficult to solve the nonlinear equation det J(q) = 0. Therefore, we now introduce the method of decoupling singularities, which is applicable whenever, for example, the manipulator is equipped with a spherical wrist.

4.9.1 Decoupling of Singularities

We saw in Chapter 3 that a set of forward kinematic equations can be derived for any manipulator by attaching a coordinate frame rigidly to each link in any manner that we choose, computing a set of homogeneous transformations relating the coordinate frames, and multiplying them together as needed. The DH convention is merely a systematic way to do this. Although the resulting equations are dependent on the coordinate frames chosen, the manipulator con-figurations themselves are geometric quantities, independent of the frames used to describe them. Recognizing this fact allows us to decouple the determination of singular configurations, for those manipulators with spherical wrists, into two simpler problems. The first is to determine so-called arm singularities, that is, singularities resulting from motion of the arm, which consists of the first three or more links, while the second is to determine the wrist singularities resulting from motion of the spherical wrist.

For the sake of argument, suppose that n = 6, that is, the manipulator consists of a 3-DOF arm with a 3-DOF spherical wrist. In this case the Jacobian is a 6 ⇥ 6 matrix and a configuration q is singular if and only if

det J(q) = 0 (4.110)

If we now partition the Jacobian J into 3 ⇥ 3 blocks as

J = [JP | J^O] =

JO = 

z3⇥ (o⁶ o3) z4⇥ (o⁶ o4) z5⇥ (o⁶ o5)

z3 z4 z5 (4.112)

Since the wrist axes intersect at a common point o, if we choose the coordi-nate frames so that o3= o4= o5= o6= o, then JO becomes

JO =  0 0 0

z3 z4 z5 (4.113)

In this case the Jacobian matrix has the block triangular form

J = 

J11 0

J21 J22 (4.114)

with determinant

det J = det J11det J22 (4.115) where J11and J22are each 3⇥3 matrices. J11has i-th column zi 1⇥ (o o^{i 1}) if joint i is revolute, and zi 1if joint i is prismatic, while

J22 = [z3 z4 z5] (4.116)

Therefore the set of singular configurations of the manipulator is the union of the set of arm configurations satisfying det J11 = 0 and the set of wrist configurations satisfying det J22= 0. Note that this form of the Jacobian does not necessarily give the correct relation between the velocity of the end-e↵ector and the joint velocities. It is intended only to simplify the determination of singularities.

4.9.2 Wrist Singularities

We can now see from Equation (4.116) that a spherical wrist is in a singular configuration whenever the vectors z3, z4and z5are linearly dependent. Refer-ring to Figure 4.3 we see that this happens when the joint axes z3 and z5 are collinear. In fact, when any two revolute joint axes are collinear a singularity results, since an equal and opposite rotation about the axes results in no net motion of the end-e↵ector. This is the only singularity of the spherical wrist, and is unavoidable without imposing mechanical limits on the wrist design to restrict its motion in such a way that z3 and z5 are prevented from lining up.

4.9.3 Arm Singularities

To investigate arm singularities we need only to compute J11, which is done using Equation (4.77) but with the wrist center o in place of on.

SINGULARITIES 135

z

₄

6 4

5

= 0

z

3

z

5

Fig. 4.3 Spherical wrist singularity.

z

2

x

0

z

0

x

1

x

2

z

1

y

1

y

2

y

0

O

c

d

⁰_c

Fig. 4.4 Elbow manipulator.

Example 4.9 Elbow Manipulator Singularities

Consider the three-link articulated manipulator with coordinate frames at-tached as shown in Figure 4.4. It is left as an exercise (Problem 4-14) to show that

J11= 2

4 a2s1c2 a3s1c23 a2s2c1 a3s23c1 a3c1s23

a2c1c2+ a3c1c23 a2s1s2 a3s1s23 a3s1s23

0 a2c2+ a3c23 a3c23

3

5 (4.117)

and that the determinant of J11 is

det J11 = a2a3s3(a2c2+ a3c23). (4.118) We see from Equation (4.118) that the elbow manipulator is in a singular configuration whenever

s3 = 0, that is, ✓3= 0 or ⇡ (4.119) and whenever

a2c2+ a3c23 = 0 (4.120)

3

= 0

³

= 180

Fig. 4.5 Elbow singularities of the elbow manipulator.

The situation of Equation (4.119) is shown in Figure 4.5 and arises when the elbow is fully extended or fully retracted as shown. The second situation of Equation (4.120) is shown in Figure 4.6. This configuration occurs when

z

0 1

Fig. 4.6 Singularity of the elbow manipulator with no o↵sets.

the wrist center intersects the axis of the base rotation, z0. As we saw in Chapter 3, there are infinitely many singular configurations and infinitely many solutions to the inverse position kinematics when the wrist center is along this axis. For an elbow manipulator with an o↵set, as shown in Figure 4.7, the wrist center cannot intersect z0, which corroborates our earlier statement that points reachable at singular configurations may not be reachable under arbitrarily small perturbations of the manipulator parameters, in this case an o↵set in either the elbow or the shoulder.

⇧

SINGULARITIES 137

z

0

d

Fig. 4.7 Elbow manipulator with shoulder o↵set.

Example 4.10 Spherical Manipulator

Consider the spherical arm of Figure 4.8. This manipulator is in a singular

1

z

0

Fig. 4.8 Singularity of spherical manipulator with no o↵sets.

configuration when the wrist center intersects z0 as shown since, as before, any rotation about the base leaves this point fixed.

⇧

Example 4.11 SCARA Manipulator

We have already derived the complete Jacobian for the the SCARA manip-ulator. This Jacobian is simple enough to be used directly rather than deriving the modified Jacobian as we have done above. Referring to Figure 4.9 we can see geometrically that the only singularity of the SCARA arm is when the elbow is fully extended or fully retracted. Indeed, since the portion of the Jacobian of the SCARA governing arm singularities is given as

J11 = 2

4 ↵1 ↵3 0

↵2 ↵4 0

0 0 1

3

5 (4.121)

where

↵1 = a1s1 a2s12 (4.122)

↵2 = a1c1+ a2c12

↵3 = a1s12

↵4 = a1c12 (4.123)

INVERSE VELOCITY AND ACCELERATION 139

z

0

z

₁

z

₂

2

= 0

Fig. 4.9 SCARA manipulator singularity.

we see that the rank of J11 will be less than three precisely whenever ↵1↵4

↵2↵3= 0. It is easy to compute this quantity and show that it is equivalent to (Problem 4-16)

s2= 0, which implies ✓2= 0, ⇡. (4.124)

⇧

4.10 INVERSE VELOCITY AND ACCELERATION

In document EL LADO POSITIVO DE LA PSICOPATÍA (página 56-62)