Robot Modeling and Control. Mark W. Spong

a two-link planar robot. Each coordinate frame moves as the corresponding link moves. The mathematical description of the robot motion is thus reduced to a mathematical description of moving coordinate frames.

(1.1)

(1.2)

      in which a₁ and a₂ are the lengths of the two links, respectively. Also the orientation of the tool frame relative to the base frame is given by the direction cosines of the x₂ and y₂ axes relative to the x₀ and y₀ axes, that is,

      (1.3)

      which we may combine into a rotation matrix

(1.4)

Equations (1.1), (1.2), and (1.4) are called the forward kinematic equations for this arm. For a six-DOF robot these equations are quite complex and cannot be written down as easily as for the two-link manipulator. The general procedure that we discuss in Chapter 3 establishes coordinate frames at each joint and allows one to transform systematically among these frames using matrix transformations. The procedure that we use is referred to as the Denavit–Hartenberg convention. We then use homogeneous coordinates and homogeneous transformations, developed in Chapter 2, to simplify the transformation among coordinate frames.

      Chapter 4: Velocity Kinematics

      To follow a contour at constant velocity, or at any prescribed velocity, we must know the relationship between the tool velocity and the joint velocities. In this case we can differentiate Equations (1.1) and (1.2) to obtain

      (1.5)

      Using the vector notation

and

, we may write these equations as

(1.6)

      The matrix J defined by Equation (1.6) is called the Jacobian of the manipulator and is a fundamental object to determine for any manipulator. In Chapter 4 we present a systematic procedure for deriving the manipulator Jacobian.

      The determination of the joint velocities from the end-effector velocities is conceptually simple since the velocity relationship is linear. Thus, the joint velocities are found from the end-effector velocities via the inverse Jacobian

      (1.7)

      where J^{− 1} is given by

The determinant of the Jacobian in Equation (1.6) is equal to a₁a₂sin θ₂. Therefore, this Jacobian does not have an inverse when θ₂ = 0 or θ₂ = π, in which case the manipulator is said to be in a singular configuration, such as shown in Figure 1.20 for θ₂ = 0. The determination of such singular configurations is important for several reasons. At singular configurations there are infinitesimal motions that are unachievable; that is, the manipulator end effector cannot move in certain directions. In the above example the end effector cannot move in the positive x₂ direction when θ₂ = 0. Singular configurations are also related to the nonuniqueness of solutions of the inverse kinematics. For example, for a given end-effector position of the two-link planar manipulator, there are in general two possible solutions to the inverse kinematics. Note that a singular configuration separates these two solutions in the sense that the manipulator cannot go from one to the other without passing through a singularity. For many applications it is important to plan manipulator motions in such a way that singular configurations are avoided.

Figure 1.20 A singular configuration results when the elbow is straight. In this configuration the two-link robot has only one DOF.

      Chapter 5: Inverse Kinematics

Now, given the joint angles θ₁, θ₂ we can determine the end-effector coordinates x and y from Equations (1.1) and (1.2). In order to command the robot to move to location A we need the inverse; that is, we need to solve for the joint variables θ₁, θ₂ in terms of the x and y coordinates of A. This is the problem of inverse kinematics. Since the forward kinematic equations are nonlinear, a solution may not be easy to find, nor is there a unique solution in general. We can see in the case of a two-link planar mechanism that there may be no solution, for example if the given (x, y) coordinates are out of reach of the manipulator. If the given (x, y) coordinates are within the manipulator’s reach there may be two solutions as shown in Figure 1.21,

Figure 1.21 The two-link elbow robot has two solutions to the inverse kinematics except at singular configurations, the elbow up solution and the elbow down solution.

the so-called elbow up and elbow down configurations, or there may be exactly one solution if the manipulator must be fully extended to reach the point. There may even be an infinite number of solutions in some cases (Problem 1–19).

Consider the diagram of Figure 1.22. Using the law of cosines¹ we see that the angle θ₂