Graphics Reference
In-Depth Information
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FIGURE 5.22
Inverse of the Jacobian with control term solution for a two-dimensional three-link armature of lengths 15, 10, and
5. The initial pose is {p/8, p/4, p/4} and goal is {
20, 5}. Panels show frames 0, 5, 10, 15, and 20 of a 21-frame
sequence in which the end effector tracks a linearly interpolated path to goal. All joints are biased to 0; the top row
uses gains of {0.1, 0.5, 0.1} and the bottom row uses gains of {0.1, 0.1, 0.5}.
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FIGURE 5.23
Inverse of the Jacobian solution formulated to pull the goal toward the end effector for a two-dimensional three-
link armature of lengths 15, 10, and 5. The initial pose is {p/8, p/4, p/4} and goal is {
20, 5}. Panels show frames
0, 5, 10, 15, and 20 of a 21-frame sequence in which the end effector tracks a linearly interpolated path to goal.
Avoiding the inverse: using the transpose of the Jacobian
Solving the linear equations using the pseudoinverse of the Jacobian is essentially determining the
weights needed to form the desired velocity vector from the instantaneous change vectors. An alter-
native way of determining the contribution of each instantaneous change vector is to form its projection
onto the end effector velocity vector [ 5 ]. This entails forming the dot product between the instantaneous
change vector and the velocity vector. Putting this into matrix form, the vector of joint parameter
changes is formed by multiplying the transpose of the Jacobian times the velocity vector and using
a scaled version of this as the joint parameter change vector.
y ΒΌ a J T V
Figure 5.24 shows frames from a sequence produced using the transpose of the Jacobian. Notice
how the end effector tracks to the goal position while producing reasonable interior joint angles.
 
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