Title
Contiuously Variable Transmission for Serial Link Cobot Architectures

Abstract
There are robotic applications in which the human user works in direct physical contact with the robot. Examples include surgical robotics, haptic display, and trajectory enhancing systems. Because safety and stability are of major concern in these systems, passive robots, which are intrinsically safe and stable, are attractive solutions. A cobot is a new type of passive robotic device created for use in these applications. Cobots have no joint actuators. Instead, they employ nonholonomic joints that redirect undesirable user motions rather than fight them.

The scope of this thesis is the creation and study of a nonholonomic joint called a continuously variable transmission (CVT). The CVT will be used to create cobots with serial link architectures. A cobot with n links uses n-1 CVTs to couple the rotations of each consecutive pair of joints. Each CVT is computer controlled to set the angular velocity ratio of the two joints that are coupled to it. With all the cobot's joints set to rotate in a particular ratio, its end point is constrained to a particular trajectory through space.

Cobots aid the user in task completion by providing virtual constraint surfaces that guide the user's motion. For example, a cobot could be used to constrain the motion of a surgeon's cutting tool to a trajectory that was programmed pre-operatively. Or as part of a haptic display system, a cobot could provide virtual walls for a person pushing on its end effector.

When a force on a cobot's end effector is applied perpendicular to a constraint surface, torques are created in the cobot's joints. These torques produce joint angular velocity error causing the end effector to leave the desired path and penetrate the constraint surface. At some value of applied force, the torques become too great, and the constraint surface is catastrophically violated.

In this thesis, a model of the friction forces that support joint torques is developed. From the model, equations are derived to relate joint angular velocity error to joint torque magnitude. The maximum sustainable joint torques are also calculated. I find that the relationship between joint angular velocity error and joint torque is a function of the angular velocity ratio of the joints. Experiments have been done to test the friction model and to yield additional information on CVT dynamics and resulting cobot performance. The results are used to determine what design changes can engender the creation of robust constraints.

Source: Masters Thesis, Northwestern University, March 1997

Title
Passive robots and haptic displays based on nonholonomic elements

Abstract
We describe a new architecture for passive robots and haptic displays, which we call a programmable constraint machine (PCM). An n- dofPCM can, under computer control, exhibit constraints (smooth, impenetrable virtual surfaces of dimensionality < n), or it can allow free n-dof motion.

At the heart of the PCM is a nonholonomic element, which is used as a continuously variable transmission (CVT). A rolling wheel, forinstance, can be used as a CVT. A prototype 2-dof cartesian PCM has been built, using a single rolling wheel. We sketch PCMs of higher dimensionality.

A rolling wheel may be thought of as a translational CVT, coupling the x and y velocities of its center by a transmission ratio which is thetangent of its steering angle, Its utility in a cartesian PCM motivates interest in a rotational analog for revolute architectures. We develop anovel rotational CVT which couples two angular velocities by an adjustable ratio.

Source: Proceedings of the IEEE 1996 International Conference on Robotics and Automation, Philadelphia