In the literature, the most common uses of CGA kinematic methods encompass the perception- action cycle: image and motion capture and robotic articulation. We have described a method for determining joint positions and rotations of a chain of revolute joints given some target end effector position. Executing this model in real-time on a computer can be used to con- trol the motions of a real-world robot manipulator such as a grasping arm, or conversely to control movements of a virtual simulation by following motion captured points. A few other potential applications present themselves as well:
• Computational origami. As suggested in Section 2.6, a crease pattern on paper can be considered a network of revolute joints. By creating loops of revolute constraints it may be possible to increase proficiency in modelling the movements of foldable surfaces, perhaps by coupling the results with a relaxation step.
• Protein Folding. Kinematic models of proteins are sometimes used to try to find their tertiary structures – possible low energy configurations of the molecules. These chains of molecules are typically represented as chains of revolute joints (the backbone), with additional revolute side chains. Representing these chains as a series of forward and backward reaching circles may offer new computational advantages.
• Structural Engineering. Stadium roofs, arterial stents, satellite dishes, and mobile habitats, are all examples of structures which are sometimes designed to transform between two distinct configurations: compactly packaged folded state and an efficiently opened operational unfolded state.
These categories of articulating spaces are sometimes calleddeployable structures. While the term tends to refer to aerospace applications, where the need for transportable payloads which can be opened automatically is prevalent, the same geometries of folding can be applied
found in nature, including twisting and untwisting DNA strands, unfurling moth tongues, opening insect wings, and inflating lungs [20, 85]. Indeed, Miura’s pattern from Section 2.6 is said to have been inspired by the herringbone pattern of an unfolding hornbeam leaf [103]. Spatial mechanisms such as the Bennett mechanism, in which the axes of the revolute joints are not parallel, are more complicated than planar mechanisms, in which the move- ment is restricted to the plane.3 As You and Chen explain in Motion Structures, the vast majority of linkage designs on this planet are planar mechanisms, meaning they do not artic- ulate in 3D space; their paths of motion are restricted to a 2D plane. 3D articulations can then be constructed by combining articulations in two directions, as in the case of the Hoberman sphere. Similarly, Guest, in his 1994 PhD thesis on the deployment of cylinders and mem- branes, explains that most space structures are 1 or 2 dimensional – either articulating beams or struts, or else solar sails or arrays and antennae.4
In [22], Chen and You discuss the under-utilized 4-bar, 5-bar, and 6-bar spatial linkage mechanisms, of which there are 15 (and growing), and propose their increased use in deploy- able structure design. In particular, in their book Motion Structures, You and Chen investigate the many forms that can be created using a network of Bennett linkages [138], some exam- ples of which we will generate in this section. The Bennett mechanism can be linked to form networks of motion, where the movement is itself the target of realization.
The use of closed loop linkages for the construction of deployable structures was first investigated by Gan and Pellegrino in [56], though as they mention, they found previous work in Crawford, R. F., Hedgepeth, J. M. and Preiswerk, P. R. (1973). In [109], Pellegrino and Vincent consider the problem of packaging flat membranes.
3Configurations of planar linkages are investigated by Demaine in his dissertation [35].
4In aerospace, the complexity of designing 3 dimensional deployable forms is canonized by the challenge
of building parabolic reflectors. Satellite dishes must minimize their volumes in the folded state and maximize their precision in the unfolded state. Guest credits Huso, Lanford, and Scheel with developing some early contributions to the notion of wrapping a flat membrane around a central hub. In the Space Structures Laboratory at the California Institute of Technology, Sergo Pellegrino has led research investigating various deployment mechanisms for microgravity conditions.
a) c)
b) d)
Figure 4.13: Linked Bennett mechanisms after You and Chen [138], which move with one degree-of-freedom. a) Profile view. b) Top View. c) By shifting the connection point between consecutive linkages, the configuration can be used to generate a circular truss. d) A twisted truss.
Figure 4.14: Deployment sequence of a circular truss composed of linked Bennett mecha- nisms.
Figure 4.15: Transformation sequence of a linked Bennett mechanism, twisted in one direc- tion, and circular in the other.
a)
b)
Figure 4.16: Deployment sequence of a proposed structure constructed with linked Bennett mechanisms: circular in one direction and twisted in the other. a) Top and b) side view.
4.6 Discussion and Future Work
We have demonstrated how the motor algebra of the conformal model can be used in conjunc- tion with the coincidence of round elements to construct complex articulations that typically do not receive such unified treatment. By working within a homogeneous representation of 3D Euclidean space that admits and organizes all its isometric transformations, we were able to avoid notational complications that emerge from compounding trigonometric functions. It is apparent from these initial studies that the use of spheres to represent isometric con- straints can be employed to find legitimate configuration spaces of overconstrained systems. It remains to be shown whether these construction methods are more precise.
We have been able to apply the Denavit-Hartenberg parameters for linkage mechanisms [36], while avoiding their matrix formulations, and detailed a strategy for encoding these parameters in terms of motors based on the work of Bayro-Corrochano. Illustrating the ge- ometric representation of constraints they elicit, we applied this representation to an imple- mentation of the approach suggested by the FABRIK method of Aristidou and Lasenby, and constructed a detailed algorithm for the inverse kinematics of a chain consisting of an arbi- trary number of revolute joints (collision detection between joints was not addressed). This also required specifying expressions for extracting the circle of revolution given a set of link parameters. In contrast to this iterative approach, we then constructed a closed-form algo- rithm for the modelling of Bennett linkages, and then, inspired by You and Chen [138], we presented some forms that can be made by linking them together.
The potential applications of this work combine the motor algebra with the geometric construction capabilities in order to fabricate objects. It would be interesting to see gear systems and machinery in general designed with the constructivist approach applied here. A good resource for finding more linkage problems to cast into GA terms is Phillips [114]. To accomplish this, collision testing algorithms need to be developed to prevent self-crossing during both forward and inverse kinematic solutions. To further the treatment of kinematics,
investigations into the modelling of higher pair joints are necessary, where the motion screws are not invariant with respect to the points of contact between links. For these contact-based models, more details are required in order to fully express the geometric constraints at play in terms of geometric algebra.
Chapter 5
Transformations: Constructions through
Interpolation of the Bivector Exponential
Jo sóc geòmetra, que vol dir sintètic.1
-Antonio Gaudí
5.1 Summary
Here we more closely consider the synthesis techniques allowed by the representation of transformations as the exponentiation of bivectors. Continuing from the previous chapter, we begin with motors as exponentials of dual lines, and the interpolations that this representation allows. We will demonstrate the use of dual lines to build trilinearly interpolated twistdefor- mation fields and then extend these methods into exponentiation of point pairs in preparation for the final Chapter. Using point pairs as exponentials, we introduce the action of rotation around a circle, for which the screws and twists serve as a special case. We show how this can be formulated into toroidal knots following the work of Dorst and Valkenburg in [46], and find a connection between that formulation and the Hopf fibration.
In [135], Wareham and Lasenby propose the use of linear and quadratic interpolations of bivector-valued rotor logarithms to blend frame positions, a subject also explored in [134], where the method is applied to deformation of a mesh. Those papers also contain algorithms to exponentialize dual line bivector “twists” into motors and conversely to find the logarithm. Belón extends those results to create more versatile mesh deformations in [13]. Sommer, Rosenhahn, and Perwass formulate “coupled twists” in [127] to synthesize forms. This is taken up by Perwass more completely in [111]. In [46], Dorst and Valkenburg give a rigorous treatment of the general bivector logarithm and reveal how it can be used to createconformal orbits, including knots. In [29], we take up the task of using these rotors to generate surfaces, a development we explore more fully in the next Chapter. Additional explorations into circle blending can be found on Ian Bell’s website [12].