In order to use a wrist-mounted sensor (such as a camera) for a robot task, the position, and orientation of the sensor with respect to the robot wrist frame must be known. We can find the sensor mounting position by impying 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 the robot wrist position, B is the resulting sensor displacement, and X is the sensor position relative to the robot wrist. The solution to an equation of this form has one degree of rotational freedom and one degree of translational freedom if the angle of rotation of A is neither 0 nor tt radians. To solve for X uniquely, it is necessary to make two arm movements and form a system of two equations of the form: A1X=XB1 and A2X=XB2. A closed-form solution to this system of equations is developed and the necessary conditions for uniqueness is stated.

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In this paper we develop an algorithm to modify the trajectories of multiple robots in cooperative manipulation. If a given trajectory results in joint torques which exceed the admissible torque range for one or more joints, the algorithm slows down or speeds up the trajectory so as to maintain all the torques within the admissible boundary. Our trajectory modification algorithm uses the concept of time scaling developed by Hollerbach[10] for single robots. A multiple robot system in cooperative manipulation has significantly different dynamics compared to single robot dynamics. As a result, time scaling algorithm for single robots is not usable with multi-robot system. The trajectory scaling schemes described in this paper requires the use of linear programming techniques and is designed to accommodate the internal force constraints and payload distribution strategies. As the multi-robot system is usually redundantly actuated, the actuator torques may be found from the quadratic minimization which has the effect of lowering energy consumption for the trajectory. A scheme for generating a robust multi-robot trajectories when the carried load mass and inertia matrix are unknown but vary within a certain range is also described in this paper. Several examples are given to show the effectiveness o f our multi-robot trajectory sealing scheme.

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In this paper we describe a methodology to determine the time-optimal trajectories for cooperative multimanipulator systems (CMMS). The dynamics of CMMS and the joint torque constraints and the constraints on the intemal forces and the distribution of the payload are incorporated into our algorithm. Linear programming techniques are utilized to find the admissible phase plane boundary. Once the admissible phase plane region is found, the actual CMMS time-optimal trajectory can be constructed by employing similar approach used in single robot minimum time trajectory planning techniques. The minimum-time trajectories which are obtained from our algorithm, for a fixed path, result in trajectories which do not apply excessive forces on the manipulated object.