This paper examines the problem of controlling a manipulator near a surface. It is assumed that the manipulator must move near or along the surface without contacting the surface. The problem, therefore, incorporates fine motion control and obstacle avoidance. It can occur in applications, such as telerobotic manipulation in space and optical inspection of surfaces. The strategy uses the kinematic constraints that arise from formulating the kinematics and dynamics in the image space of dual quaternion coordinates. Through Lagrange multipliers the constraints are incorporated directly into the dynamics and used to force the manipulator end effector to follow a path along the desired surface. The constraints appear algebraically, and can, therefore, be manipulated directly when developing and implementing strategies for avoiding collisions with the surface. The dynamics are formulated in the operational image space of the manipulator. Thus, the results are easily extended to multiple robot configurations.>

The equations of motion of cooperating robot systems are obtained by connecting the individual equations of motion for each arm and the workpiece using the constraint equations of the closed chain. Dual quaternions have been shown to provide a convenient algebraic representation for these constraints. The equations of motion for a rigid body whose position is defined by the eight dual quaternion coordinates are derived. Because a rigid body has six degrees of freedom, the use of dual quaternion coordinates requires two additional differential constraint equations. The result is a set of ten differential equations prescribing the movement of the body. Use of these equations is demonstrated through a planar example of a double pendulum.>

An examination is made of the equations for the boundaries of the joint-space obstacles in the workspace of a 5R closed-chain robot. This two-degree-of-freedom chain appears in many robotics applications. The equations describing the joint-space obstacle boundaries are derived, using three different sets of parameters, and the obstacles are graphically displayed. The boundaries of polygonal obstacles in contact with the polygonal links of a robot arm are studied using the 4D image space of planar displacements. The obstacles are mapped into the image space as the surface representing the position and orientation that a body assumes while maintaining contact with the obstacle. The intersection of the available robot positions and the obstacle contact positions is a representation of the obstacles in the robot's workspace. Three parameterizations of joint-space obstacles are determined from the image-space representation: the joint angles of one of the 2R robots, the end effector position, and the base joint angles of the 5R chain. The first and third parameter sets define two possible configurations for the 5R chain, and the second defines four possible configurations.>