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Joint Space Versus Cartesian Space. An n-link manipulator has n degrees of freedom, and the position of the end-effector is completely fixed once the joint vari-ables qi are prescribed. This position may be described either in joint coordinates or in Cartesian coordinates. The joint coordinates position of the end-effector is sim-ply given by the value of the n-vector q. The Cartesian position of the end-effector is given in terms of the base frame by specifying the orientation and translation of a coordinate frame affixed to the end-effector in terms of the base frame; this is exactly the meaning of T(q). That is, T(q) gives the Cartesian position of the end-effector. The Cartesian position of the end-effector may be completely specified in our 3-D space by a six-vector; three coordinates are needed for translation and three for orientation. The representation of Cartesian translation by the arm T(q) matrix is suitable, as it is simply given by p(q) [x y . Unfortunately, the representation of Cartesian orientation by the arm T matrix is inefficient in that R(q) has nine elements. More efficient representations are given in terms of quaternions or the tool configuration vector.
Kinematics and Inverse Kinematics Problems. The robot kinematics prob-lem is to determine the Cartesian position of the end-effector once the joint variables are given. This is accomplished simply by computing T(q) for a given value of q. The inverse kinematics problem is to determine the required joint angles qi to position the end-effector at a prescribed Cartesian position. This corresponds to solving (3.1.2) for q E SR” given a desired orientation R and translation p of the end-effector. This is not an easy problem, and may have more than one solution (e.g. think of picking up a coffee cup— one may reach with elbow up, elbow down, etc.). There are various efficient techniques for accomplishing this. One should avoid the functions arcsin, arccos and use where possible the numerically well-conditioned arctan function.
3.1.2 Robot Jacobians Kinematics transformations deal with conversion of positions between various coor-dinate frames. Jacobians allow the transformation of dynamical quantities including velocities, accelerations, and forces.
Transformation of Velocity and Acceleration. When the manipulator moves, the joint variable becomes a function of time t. Suppose there is prescribed a generally nonlinear transformation from the joint variable q(t) E Rn to another variable y(t) E RP given by
Y(t) = h(q(t))• (3.1.3) An example is provided by the equation y = T(q), where y(t) is the Cartesian position. Taking partial derivatives one obtains . ah = J(q)4, (3.1.4)
where 1(q) is the Jacobian associated with h(q). This equation tells how the joint velocities 4 are transformed to the velocity Y.
If y = T(q) the Cartesian end-effector position, then the associated Jacobian T(q) = 1(:) is known as the manipulator Jacobian. There are several techniques for efficiently computing this particular Jacobian. Note that y = c.,T1T Es, with v G ar the linear velocity and w E the angular velocity. Therefore, in formally computing 1(q) there are some complications arising from the fact that the representation of orientation in the homogeneous transformation T(q) is a 3 x 3 rotation matrix and not a 3-vector.. If the arm has n links, then the Jacobian is a 6 x n matrix; if n is less than 6 (e.g. SCARA arm), then 1(q) is not square and there is not full positioning freedom of the end-effector in 3-D space. The singularities of T(q) (where it loses rank) define the limits of the robot workspace; singularities may occur within the workspace for some arms. Another example of interest is when y(t) is the position in a camera coordinate frame. Then 1(q) reveals the relationships between manipulator joint velocities (e.g. joint incremental motions) and incremental motions in the camera image. This affords a technique, for instance, for moving the arm to cause desired relative motion of a camera and a workpiece. Nate that, according to the velocity trans-formation (3.1.4), one has that incremental motions are transformed according to = J(q)Aq• Differentiating (3.1.4) one obtains the acceleration transformation = +
Force ‘Transformation. Using the notion of virtual work, it can be shown that forces in terms of q may be transformed to forces in terms of y using r = JT (OF,
where r(t) is the force in joint space (given as an n-vector of torques for a revolute robot), and F is the force vector in y space. If y is the Cartesian position, then F is a vector of three forces Yr f,, MT and three torques [rr When 1(q) loses rank, the arm cannot exert forces in all directions that may be specified.
Robot dynamics considers motion effects due to the control inputs and inertias, Coriolis forces, gravity, disturbances, and other effects. It reveals the relation be-tween the control inputs and the joint variable motion q(t), which is required for the purpose of servo-control system design. A robot manipulator can have either revolute joints or prismatic joints. The values of the angles, for revolute joints, and link lengths, for prismatic joints, are called the link variables and are denoted q2 (t), , q, (t) for joints one, two, and so on. The number of links is denoted n; for complete freedom of motion in space, six degrees of freedom are needed, three for positioning, and three for orientation. Thus, many commercial robots have 6 links. We discuss here robots which are rigid, that is which have no flexibility in the links or in the gearing of the joints; flexible robots are discussed in Chapter 5.

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