Finding the mounting position of a sensor by solving a homogeneous transform equation of the form AX = XB

In order to use a wrist-mounted sensor (such as a camera, a range sensor, or a tactile sensor) for a robot task, the position of the sensor with respect to T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</inf> (wrist of robot) must be known. We can find the mounting position of the sensor by moving the robot and observing the resulting motion of the sensor. This yields a homogeneous transform equation of the form AX=XB, where A is the change in T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</inf> due to the arm movement, B is the resulting sensor displacement, and X is the sensor position relative to T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</inf> . A and B are known, since A can be computed from the encoder values and B can be found by the sensor system. The solution to an equation of this form has one degree of rotational freedom and one degree of translational freedom. In order to solve for X (the sensor position) uniquely, it is necessary to make two arm movements and form a system of two equations of the form: A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> X=XB <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> X=XB <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> . A closed-form solution to this system of equations is presented. The necessary condition for a unique solution is that the axes of rotation of A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> are neither parallel or antiparallel to one another. The theory is supported by simulation results.