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Abstract Kinematic synthesis to meet an approximate motion specification naturally forms a constrained optimization problem. In this work, we conduct global design searches by direct computation of all critical points through stationarity conditions. The idea is not new, but performed at scale is only possible through modern polynomial homotopy continuation solvers. Such a complete computation finds all minima, including the global, serving as a powerful design exploration technique. We form equality constrained objective functions that pertain to the synthesis of spherical four-bar linkages for approximate function generation. For each problem considered, Lagrangian stationarity conditions set up a square system of polynomials. We consider the most general case where all mechanism dimensions may vary, and a more specific case that enables the placement of ground pivots. The former optimization problem is shown to permit an estimated maximum of 268 sets of critical points, and the latter permits 61 sets. Critical points are classified as saddles or minima through a post-process eigenanalysis of the projected Hessian. Approximate motion is specified as discretized points from a desired input-output angle function. The coefficients of the stationarity polynomials can be expressed as summations of symmetric matrices indexed by the discretization points. We take the sums themselves to parameterize these polynomials rather than constituent terms (the discrete data). In this way, the algebraic structure of the synthesis equations remains invariant to the number of discretization points chosen. The results of our computational work were used to design a mechanism that coordinates the unfolding of wings for a deployable aircraft. 

A spherical five-bar linkage generally has two degrees-of-freedom. However, some configurations arise where the mechanism is able to move with an additional degree-of-freedom, even with both input angles fixed. These configurations are termed exceptional. They appear in a configuration surface as embedded straight lines in the direction of its outputs. Without prior knowledge, these lines can be discovered by structured slicing of the configuration surface using the tools of polynomial homotopy continuation. This search procedure is conducted for spherical five-bars of equal-link-length. Cases for adjacent and nonadjacent input angles are analyzed. An equal-link-length spherical five-bar with 90 deg links is used for the design of a deployable control surface for an aircraft. Taking advantage of the exceptional sets in the mechanism, one motor is able to lock, deploy, and articulate the control surface. 

Mechanisms of exceptional mobility, including both overconstrained mechanisms and robots with self-motion, move with more degrees of freedom than predicted by the Grübler–Kutzbach formulas. Although a number of such cases are known, it is difficult to find new examples. This article explains a geometric formulation, called a fiber product, that facilitates finding exceptional mechanisms using tools from numerical algebraic geometry. The purpose of this article is to specialize the mathematical theory developed in A.J. Sommese and C.W. Wampler (2008) to the realm of kinematics and to present simple planar, spherical, and spatial examples that illustrate basic concepts. Although the formulation is general, its application to more complicated mechanisms will require the development of more refined solution techniques that exploit the symmetry inherent in fiber products. 

Repetitive subtasks of locomotion are offloaded from a conventional computer-actuator-sensor set-up to automatic mechanical processes. The subtasks considered are: (1) when out-of-contact with the environment, move a leg to a ready position in preparation for step contact, and (2) when contact is detected, push off the ground. Using conventional closed-loop control, subtask (1) would be accomplished by programming logic and a feedback loop onto a computer-motor-encoder system, and subtask (2) would be accomplished by sensing contact, then commanding the leg motor to push-off via programmed computer logic. We demonstrate how to transition this programmed logic from a computer processor to a mechanical processor. The mechanical processor performs preprogrammed actions based on combinations of states of components, some of which are internal and some that interact with the environment. Because signals are not digital, but rather mechanical quantities of energy, position, and force; transitioning to a mechanical processor enables a third subtask not possible by the computer alone: that is, (3) the accumulation of elastic energy while out-of-contact with the environment, and its automatic release upon contact for a more powerful push-off motion. Migrating processing out of the computer reduces the number of transduction steps, allows for faster responses to dynamic events, and instantiates a high-powered reflex triggered by ground contact. To illustrate these benefits, a robot with built-in onboard mechanical processing is compared to a conventional robot with logic executed by an offboard computer. 

The transmission properties of multi-degree-of-freedom mechanisms can be tuned by shaping velocity ellipses throughout their workspace. Velocity ellipses are the image of a circle in the actuator velocity space mapped by the Jacobian into end-effector velocities. In this work, two machine learning methods using convolutional neural network architectures are proposed to synthesize planar 2R mechanism designs that approximately produce the desired velocity ellipses. An ensemble of image-based regression models is trained in a supervised fashion to output multiple 2R designs that approximate the specified ellipses. As an alternative to this approach, a second physics-informed neural network is constructed to train an ensemble of encoder models without specifying the 2R link lengths. During training, a decoder model that approximates the kinematics (physics) of the 2R is used to find how well the 2R design output by the encoder approximates the specified ellipses. These models are used to obtain multiple 2R designs that produce ellipses suitable for legged locomotion tasks and some preliminary results are presented. 

There are many ways for a gripper to estimate the forces between its fingers. If powered by direct-drive brushless motors, then one technique is to measure their current. This is not the most accurate technique, but it is simple, keeps the sensor remote, and requires no new components. The estimation involves multiplying current signals through by the torque constant and the inverse transpose of the Jacobian. The Jacobian either amplifies the signal from fingertip force to motor current (at the cost of tip force production), or diminishes it (with the gain of tip force production), indicating an inherent trade-off. However, the Jacobian is a function of configuration, and for any workspace point there are multiple configurations (multiple inverse kinematics solutions), therefore a selection of Jacobian exists. For a given workspace point, the number of Jacobian choices is just a few, but these choices can be designed (through dimensional synthesis) to overcome the trade-off. The problem can be framed as velocity ellipse synthesis over multiple output modes. In this work, we conduct optimal synthesis to compute a new gripper design. The gripper was built and tested. It transitions between two different modes: sense mode and grip mode. Sense mode can sense forces 3 times smaller than grip mode. Grip mode can exert forces 4 times greater than sense mode. 

Articulation between body segments of small insects and animals is a three degree-of-freedom (DOF) motion. Implementing this kind of motion in a compact robot is usually not tractable due to limitations in small actuator technologies. In this work, we concede full 3-DOF control and instead select a one degree-of-freedom curve in SO(3) to articulate segments of a caterpillar robot. The curve is approximated with a spherical four-bar, which is synthesized through optimal rigid body guidance. We specify the desired SO(3) motion using discrete task positions, then solve for candidate mechanisms by computing all roots of the stationary conditions using numerical homotopy continuation. A caterpillar robot prototype demonstrates the utility of this approach. This synthesis procedure is also used to design prolegs for the caterpillar robot. Each segment contains two DC motors and a shape memory alloy, which is used for latching and unlatching between segments. The caterpillar robot is capable of walking, steering, object manipulation, body articulation, and climbing. 

Kinematic synthesis to meet an approximate motion specification naturally forms a constrained optimization problem. Instead of using local methods, we conduct global design searches by direct computation of all critical points. The idea is not new, but performed at scale is only possible through modern polynomial homotopy continuation solvers. Such a complete computation finds all minima, including the global, which allows for a full exploration of the design space, whereas local solvers can only find the minimum nearest to an initial guess. We form equality-constrained objective functions that pertain to the synthesis of spherical four-bar linkages for approximate function generation. We consider the general case where all mechanism dimensions may vary and a more specific case that enables the placement of ground pivots. The former optimization problem is shown to permit 268 sets of critical points, and the latter permits 61 sets. Critical points are classified as saddles or minima through a post-process eigenanalysis of the projected Hessian. The discretization points are contained within the coefficients of the stationarity polynomials, so the algebraic structure of the synthesis equations remains invariant to the number of points. The results of our computational work were used to design a mechanism that coordinates the folding wings. We also use this method to parameterize mechanism dimensions for a family of hyperbolic curves. 

In this paper, we form a constrained optimization problem for spherical four-bar motion generation. Instead of using local optimization methods, all critical points are found using homotopy continuation solvers. The complete solution set provides a full view of the optimization landscape and gives the designer more freedom in selecting a mechanism. The motion generation problem admits 61 critical points, of which two must be selected for each four-bar mechanism. We sort solutions by objective value and perform a second order analysis to determine if the solution is a minimum, maximum, or saddle point. We apply our approximate synthesis technique to two applications: a hummingbird wing mechanism and a sea turtle flipper gait. Suitable mechanisms were selected from the respective solution sets and used to build physical prototypes.