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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. 

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.