We present a constrained Rigid Body Dynamics (RBD) that guarantees satisfaction of kinematic constraints, enabling direct simulation of complex mechanical systems with arbitrary kinematic structures. To ensure constraint satisfaction, we use an implicit integration scheme. For this purpose, we derive compatible dynamic equations expressed through the quaternion time derivative, adopting an additive approach to quaternion updates instead of a multiplicative one, while enforcing quaternion unit-length as a constraint. We support all joints between rigid bodies that restrict subsets of the three translational or three rotational degrees of freedom, including position- and force-based actuation. Their constraints are formulated such that Lagrange multipliers are interpretable as joint forces and torques. We discuss a unified solution strategy for systems with redundant constraints, overactuation, and passive degrees of freedom, by eliminating redundant constraints and navigating the subspaces spanned by multipliers. As our method uses a standard additive update, we can interface with unconditionally-stable implicit integrators. Moreover, the simulation can readily be made differentiable as we show with examples.

The kinematic motion of a robotic character is defined by its mechanical joints and actuators that restrict the relative motion of its rigid components. Designing robots that perform a given target motion as closely as possible with a fixed number of actuated degrees of freedom is challenging, especially for robots that form kinematic loops. In this paper, we propose a technique that simultaneously solves for optimal design and control parameters for a robotic character whose design is parameterized with configurable joints. At the technical core of our technique is an efficient solution strategy that uses dynamic programming to solve for optimal state, control, and design parameters, together with a strategy to remove redundant constraints that commonly exist in general robot assemblies with kinematic loops. We demonstrate the efficacy of our approach by either editing the design of an existing robotic character, or by optimizing the design of a new character to perform a desired motion.

This article introduces a computational approach for routing thin artificial muscle actuators through hyperelastic soft robots, in order to achieve a desired deformation behavior. Provided with a robot design and a set of example deformations, we continuously co-optimize the routing of actuators, and their actuation, to approximate example deformations as closely as possible. We introduce a data-driven model for McKibben muscles, modeling their contraction behavior when embedded in a silicone elastomer matrix. To enable the automated routing, a differentiable hyperelastic material simulation is presented. Because standard finite elements are not differentiable at element boundaries, we implement a moving least squares formulation, making the deformation gradient twice differentiable. Our robots are fabricated in a two-step molding process, with the complex mold design steps automated. While most soft robotic designs utilize bending, we study the use of our technique in approximating twisting deformations on a bar example. To demonstrate the efficacy of our technique in soft robotic design, we show a continuum robot, a tentacle, and a four-legged walking robot.