The combination of manipulator arm and mobile base introduces redundancy in the process of manipulation for DM^2. Thus, there are virtually infinite ways to coordinate the position of the mobile base and the end-effector so that the end effector is at a specified point in the robot's workspace. To take advantage of this redundancy, we have developed a coordination scheme for the base and manipulator arm which can optimize the use of both with respect to a set of performance criteria. These criteria are (1) the manipulability of the manipulator arm, (2) the distance the base must move, and (3) the static stability of the whole robot when it takes on a load. The result of optimizing the robot's configuration for these criteria is that the robot is able to perform tasks with maximum dexterity and minimum energy consumption, while enabling it to handle loads with minimal danger of overturning. The coordination scheme we have developed allows these three criteria to be weighted with respect to one another depending on which factors are considered most important at a given time.

The first criterion, manipulability, is a common measure of a manipulator's ability to move its end effector from a given configuration. By maximizing this measure, we help to ensure that the manipulator is in a configuration that is well suited for generating the necessary motions for performing its tasks. To generate a manipulability index for DM^2, we first derive the Jacobian matrix of the manipulator from its forward kinematic equations. The scalar index of manipulability is the norm of the derivative of this Jacobian matrix. For this measure, we consider only the the effects of the three joints nearest the base of the manipulator so that the robot may use the remaining two joints to control the orientation of the gripper at the desired location.

In addition to maximizing manipulability, it is important that DM^2 conserve energy by minimizing unnecessary motion of its mobile base. Because the size and mass of the mobile base is much greater than that of manipulator arm, driving the base requires significantly more energy than moving the manipulator to generate a the same motion at the end effector. We thus measure consider the straight-line distance that the base must move an index of energy consumption required to achieve a given configuration, and attempt to minimize this index.

The last criterion we currently consider is static stability. It is important that DM^2 never try to lift a load from or move a load to any configuration for which it is in danger of overturning. We can minimize the possibility of destabilizing the robot during its work by keeping the center of the mass of the whole system as close as possible to where it was without the load. We thus use the horizontal distance of the center of mass of the loaded arm from the original center of mass as an index to be minimized.

We form an optimization function by multiplying the expressions for each of the three indices by weighting coefficients, and then subtracting resulting the energy-index term and the stability-index term from the manipulability-index term. If the weight of the load and the desired location and orientation of the end-effector are given as inputs, we can quickly maximize the optimization function to obtain a configuration of the base and manipulator that is optimal with respect to manipulability, energy consumption, and robot stability.

Christopher Lee (chrislee@ri.cmu.edu)