Parallel mechanisms have recently been targeted for the design and development of applications for real-world industrial problems. In situations where the need for accuracy and sturdiness dominate the requirement of a large workspace, parallel mechanisms present themselves as feasible alternatives to their serial counterparts. They tend to be more accurate and repeatable than serial mechanisms in part due to the fact that their position errors do not accumulate, thus the total position error rarely exceeds the error caused by a single leg of the mechanism.

Parallel mechanisms generally comprise two platforms which are connected by a plurality of prismatic joints or legs acting in-parallel. The most common configuration comprises six legs, and the legs are linear actuators such as hydraulic cylinders, or in the case of a passive mechanism they could be spring loaded, for example. One of the platforms is defined as the "movable platform," which has six degrees of freedom relative to the other platform, which is the "base." With six degrees of freedom, the movable platform is capable of moving in three linear directions and three angular directions singularly or in any combination. One such platform known as a "Stewart" platform was introduced in 1965 by D. Stewart [2] for use in an aircraft simulator. The Stewart-type parallel mechanisms are defined as those platforms whose six legs meet pair-wise in three points in at least one platform. Stewart's two suggested arrangements of the 3-3 and 6-3 Platform are shown in Figure 1;

the nomenclature signifying the points of connection on the base and the movable platform respectively. Since then, various applications of Stewart type platforms have been suggested, ranging from manipulators to force/torque sensors.

When the locations of the connection points of the prismatic legs are known for both platforms, and when the legs can be actuated and their length determined, then closed-loop control of the position and orientation of the movable platform relative to the base platform is possible, and depends on the solutions to two geometrical computations. First, when a desired position and orientation of the movable platform is known, then the corresponding desired lengths of the six legs are determined by "reverse displacement analysis." On the other hand, when the actual leg lengths are known, the determination of the actual position and orientation of the movable platform is done by "forward displacement analysis." Because there are multiple mathematical solutions to this problem, it is clear that the forward displacement analysis is far more complicated than the reverse analysis.

Because of the complexities of such parallel devices, earlier methodologies involved an iterative forward displacement analysis together with the simpler reverse displacement analysis for closed-loop control of the position and orientation of the movable platform. An iterative technique for accomplishing the forward displacement analysis has many drawbacks, such as the requirement of an undetermined amount of time for computation, dependence on a good initial guess of the position and orientation of the movable platform, and finally, the iterative solution has no way of determining regions of mechanism singularities; thus rendering no mathematical means for determining closures and leaving mechanisms susceptible to static instabilities.

On the other hand, control by a "closed-form forward displacement analysis" greatly differs from an iterative forward displacement analysis because the computation is reduced to a single polynomial in a single variable. The control method is considered "closed-form" by kinematicians because the polynomial is obtained directly from known dimensions, even though the polynomial may be rooted in an iterative way with single dimensional search. This methodology not only provides a computer-time repeatable and dependable process for determining position and orientation of the platform but also yields much important information on the geometry and kinematics of a parallel mechanism. Furthermore, control with a closed-form forward displacement controller provides a Cartesian controller with necessary feedback information, namely, the position and the orientation of the movable platform relative to the base. This is especially important when the actual position and orientation cannot be directly sensed, and when the manipulator's configuration is determined solely from lengths of the connecting prismatic legs [1].

Also, in the area of force control, a closed-form forward displacement analysis provides the necessary analytics to enhance the use of the platform as a force/torque sensor. A Stewart platform design that is based on an in-parallel structure lends itself well to static force analysis, particularly when utilizing the theory of screws. A wrench applied to the movable platform can be statically equated to the summation of forces measured along the lines of the six prismatic legs. Employing a forward displacement analysis provides the analytics to monitor gross deflections of a Stewart platform and permits the design of a more compliant force/torque sensor i.e., a controller programmed to effect a forward displacement analysis generates the geometry of the lines of the six connecting prismatic legs of the Stewart platform so that the effects of finite changes in leg lengths can be related to the forces and torques applied to the movable platform [1].

Griffis and Duffy [1] have invented a method and apparatus that incorporates closed-form forward displacement analyses into the closed-loop Cartesian control of a general class of parallel mechanisms. The inventions are suitable for use in flight simulator motion platforms, heavy object lifters, passive force/torque sensors, and other systems using parallel mechanisms. In particular, the inventions are useful in applications requiring real-time motion control. Also, a methodology has been presented, operative in real time, to carry out a method for solving a closed-form forward displacement computation, instead of an iterative computation. The closed-form forward displacement control procedure comprises transducers for acquiring the lengths of lines extending between the movable platform and the base platform, which are used to compute at least one closed-form forward displacement solution which in turn determines the position and orientation of the movable platform relative to the base platform.

The Special 6-6 Platform

Griffis and Duffy have also discovered a new, special 6-6 parallel mechanism, shown in Figure 2, whose geometry is simple so that the closed-form solution described is applicable using a geometric reduction. The novel 6-6 mechanism is capable of motion in six degrees of freedom, and comprises a base platform comprising at least three points defining a base plane, and a movable platform comprises at least three points defining a movable platform. Six linearly extensible support members extend between the base platform and the movable platform. Each of the support members are actuateable for a motion-producing system or alternatively are passively-force resistant for a motion sensing system.

Thus, the 6-6 Platform and the associated control system facilitates controlling the position and orientation of the platform of a parallel mechanism by real-time computation of a closed-form forward solution to the geometry of the platform, based on the sensed lengths of linear actuators, instead of by iterative and computation-intensive numerical methods. This capability makes this system a prime candidate for applications requiring real-time feedback and control.

As stated previously, it is possible to perform numerical iterations to obtain the position and orientation of the platform, however, such iterative solutions have a tendency to "jump" from one closure to another, an undesirable property from a practical viewpoint. The need for a closed-form analysis really manifests itself whenever a platform is to be used to control force and motion simultaneously. This need becomes more apparent in applications requiring real-time response. Also, employing a closed-form forward displacement analysis provides the analytics to consider the design of a compliant force/torque sensor.

Thus, to control force and motion simultaneously in real-time, it is necessary to perform the closed-form forward displacement analysis in real-time. There is a predicament, however. The geometrically simplest mechanism is the 3-3 Platform (an octrahedron), whose closed-form forward displacement analysis involves the solution of an eighth degree polynomial [3]. It is practical to solve such a polynomial rapidly in real time. However, such a platform has a serious mechanical disadvantage i.e., it is not possible to design the concentric ball and socket joints at each of the double connection points without mechanical interference. From a mechanical point of view, it is preferable to separate the double connection joints in order to overcome this problem resulting in a 6-6 Platform. Nevertheless, the closed-form forward solution for the general 6-6 Platform involves a fortieth degree polynomial (Wen and Liang [4] and Zhang and Song [5]); an impractical polynomial to solve in real-time.

The special 6-6 Platform incorporates the advantages of both the 3-3 Platform and the general 6-6 Platform. There are six distinct connecting points in both the top platform and the base which avoids the mechanical interference problem. At the same time, it is possible to control the position of the top platform employing the eighth degree polynomial for the forward solution of the octrahedron. This combination makes possible the simultaneous control of force and motion using this device.

A device employing the above mentioned techniques, known as the Smart Kinestatic Interactive Platform (SKIP) [6], has been designed and built at the Center for Intelligent Machines and Robotics, University of Florida. Another version of the device, to be used in an end-effector and required to fit a volume of approximately 4 cu. in., has also been designed. The applications for these mechanisms vary from assembly robots requiring large forces to micro manipulators requiring small, high-precision force-controlled motion.

1. Griffis, M. W. and Duffy, J., "Method and apparatus for controlling geometrically simple parallel mechanisms with distinctive connections," United States Patent #5179525, January 12, 1993.

2. D. Stewart, "A platform with six degrees of freedom," Proc. Inst. Mech. Eng., London, Volume 180, 1965, pp. 371-386.

3. Griffis, M. W. and Duffy, J., "A forward displacement analysis of a class of Stewart platforms," Journal of Robotic Systems, John Wiley, 6(6), 1989, pp 703-720.

4. Wen, F. A. and Liang, C.G., "Displacement analysis for the general Stewart Platform type mechanism," Beijing University of Posts and Telecommunication, private communication, June 1992.

5. Zhang, C., and Song, S., "Forward position analysis of nearly general Stewart Platforms," Proceedings of the 22nd Biannual ASME Mechanisms Conference, Scottsdale, Az, September 1992.

6. Griffis, M. W. and Duffy, J., "A smart Kinestatic Interactive Platform," Sixth Annual Conference on Recent Advances in Robotics, Gainesville, FL, April 1993.