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    A homogeneous matrix maps frame K coordinates to (k-1) coordinates. Four fundamental operations are involved in making (k-1) frame coincident with k frame.

    Given the values of joint variables Q1, Q2, … Qn, you need to solve for the end-effector location (i.e. position and orientation ) in the Cartesian space of the robot base frame.

    The 3X3 submatrix R(q) represents the tool orientation, 3X1 submatrix P(Q) represents position of the tool. The three columns of R represents the direction of unit vectors of the tool frame WRT base frame.

    Finding the position and the orientation of the tool at the soft home position of, for instance, the six-axis articulated arm, Intelledex 660T, involves the following steps:
    •Assigning the Coordinate frame
    •Getting the DH parameters
    •Getting the Arm matrix

    Manipulator tasks are normally formulated in terms of the desired position and orientation.

    A systematic closed form solution applicable to robots in general is not available. Unique solutions are rare and multiple solutions exist

    An inverse problem is more difficult than a forward problem.

    The Arm matrix represents the position P and orientation R of the tool in base coordinate frame as a function of joint variable Q

    A manipulator is solvable if all the sets of joints variables can be found corresponding to a given end-effector location. The necessary conditions are the following:
    •Tool point within the workspace
    •N ≥ 6, to have any arbitrary orientation of tool
    •Tool orientation is such that none of the joint limitations are violated

    There are two kinds of solutions:
    •Closed form solutions: analytical expressions
    •Numerical solutions: iterative search (time consuming)

    In order to get closed-form solutions, there is a sufficiency condition: three adjacent joint axes intersecting or three adjacent joint axes parallel to one another.

    There are also unique solution (multiple solution), which is the case of redundant robot. Elbow-up and elbow-down solutions exists.

    There are two approaches for deriving closed form solutions:
    •Algebraic versus geometric
    •Algebraic approach