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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. 

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. 

The kinematic configuration space of a manipulator determines the set of all possible motions that may occur, and its differential properties have a strong, albeit indirect, influence on both static and dynamic performance. By viewing first-order kinematics as a field of Jacobian-defined ellipses across a workspace, a novel two degree-of-freedom manipulator was designed, and is tested in this paper for its benefits. The manipulator exhibits a field of ellipses that biases transmission characteristics in Cartesian directions of the end-effector. The horizontal direction is biased toward speed in order to move across the width of the workspace quickly, while the vertical direction is biased toward force production in order to resist gravitational loads. The latter bias endows the manipulator with load capacity in the absence of gears. Such an exclusion can forego the extra weight, complexity, backlash, transmission losses, and fragility of gearboxes. Additionally, a direct drive set-up improves backdrivability and transparency. The latter is relevant to applications that involve interacting with the environment or people. Our novel design is set through an array of theoretical and experimental performance studies in comparison to a conventional direct drive manipulator. The experimental results showed a 3.75× increase in payload capacity, a 2× increase in dynamic tracking accuracy, a 2.07× increase in dynamic cycling frequency, and at least a 3.70× reduction in power consumption, considering both static and dynamic experiments. 

The multidirectional transmission characteristics of a five-bar linkage can be visualized by plotting Jacobian-defined velocity ellipses inside its workspace. The orientation, size, and aspect ratio of these ellipses indicate directional force and velocity multiplication from the actuators to the end-effector. Our broader goal is approximate dimensional synthesis to achieve desired ellipses. On a workspace bound, the minor axis of a velocity ellipse collapses while the major axis aligns tangential to the bound. Interior to the workspace, ellipses vary with continuity. Therefore, the shape of a workspace bound influences the interior ellipses. The workspace bounds of a five-bar linkage are formed from segment of four-bar coupler curves (the locus of endpoint positions while the five-bar is held in output singularity conditions) and circular segments. Therefore, interior ellipses can be influenced by the path synthesis of four-bar linkages that represent the five-bar situated with certain links held colinear (the output singularity conditions). This paper details the synthesis of these four-bar coupler curves for forming the workspace bounds of a five-bar in order to influence its interior ellipses. Our approach employs saddle graphs that detail the connectivity of critical points over an optimization function. 

The multidirectional transmission characteristics of a five-bar linkage can be visualized by plotting Jacobian-defined velocity ellipses inside its workspace. The orientation, size, and aspect ratio of these ellipses indicate directional force and velocity multiplication from the actuators to the end-effector. Our broader goal is approximate dimensional synthesis to achieve desired ellipses. On a workspace bound, the minor axis of a velocity ellipse collapses while the major axis aligns tangential to the bound. Interior to the workspace, ellipses vary with continuity. Therefore, the shape of a workspace bound influences the interior ellipses. The workspace bounds of a five-bar linkage are formed from segments of four-bar coupler curves (the locus of endpoint positions while the five-bar is held in output singularity conditions) and circular segments. Therefore, interior ellipses can be influenced by the path synthesis of four-bar linkages that represent the five-bar situated with certain links held colinear (the output singularity conditions). This paper details the synthesis of these four-bar coupler curves for forming the workspace bounds of a five-bar in order to influence its interior ellipses. Our approach employs saddle graphs that detail the connectivity of critical points over an optimization function. 

For a given endpoint position, a five-bar manipu-lator may assume several separate configurations, with each offering distinct differential kinematics. The corresponding configurations are separated by output singularities and are said to belong to different output modes. In this work, a procedure for dynamically switching between output modes is proposed, with each mode offering different directional force/velocity transmission ratios. The procedure involves solving an optimal control problem using a projection-based direct collocation method for constrained mechanisms to find an optimal trajectory along which the mechanism changes output modes. Using this procedure, a five-bar mechanism configured at a given end-effector position is shown to switch to another output mode where the electrical energy consumed by the actuators to statically hold the mechanism reduces by 80%. Furthermore, the computed trajectories are seen to cross input singularities, a maneuver made possible by momentum planning since actuator authority is impaired at these configurations. 

An algorithm is presented for computing the tension in an elastic cable subject to sagging under its own weight, a problem highly relevant in tethered systems such as cable-driven parallel robots. This requires solving the two coupled equations of the Irvine cable model, which give the endpoint position as a function of vertical and horizontal components of tension. Via a change of variables, we reformulate this system as a pair of uncoupled equations, which are shown to have a unique solution. We develop an efficient numerical procedure to solve one of these, after which closed-form formulas provide the solution of the second equation and ultimately the tension components. 

Designing and analyzing large cable-driven parallel robots (CDPRs) for precision tasks can be challenging, as the position kinematics are governed by kineto-statics and cable sag equations. Our aim is to find all equilibria for a given set of unstrained cable lengths using numerical continuation techniques. The Irvine sagging cable model contains both non-algebraic and multi-valued functions. The former removes the guarantee of finiteness on the number of isolated solutions, making homotopy start system construction less clear. The latter introduces branch cuts, which could lead to failures during path tracking. We reformulate the Irvine model to eliminate multi-valued functions and propose a heuristic numerical continuation method based on monodromy, removing the reliance on a start system. We demonstrate this method on an eight-cable spatial CDPR, resulting in a well-constrained non-algebraic system with 31 equations. The method is applied to four examples from literature that were previously solved in bounded regions. Our method computes the previously reported solutions along with new solutions outside those bounds much faster, showing that this numerical method enhances existing approaches for comprehensively analyzing CDPR kineto-statics.