Trade-offs in maneuverability and stability are essential in ecologically relevant situations with respect to robustness of locomotion, with multiple strategies apparent in animal model systems depending on their habitat and ecology. Free appendages such as tails and ungrounded limbs may assist in navigating this trade-off by assisting with balance, thereby increasing the acceleration that can be achieved without destabilizing the body. This comparative analysis explores the inertial mechanisms and, in some cases, fluid dynamic mechanisms by which appendages contribute to the stabilization of gait and perturbation response behaviors in a wide variety of animals. Following a broad review of examples from nature and bio-inspired robotics that illustrate the importance of appendages to the control of body orientation, two specific cases are examined through preliminary experiments: the role of arm motion in bipedal gait termination is explored using trajectory optimization, and the role of the cheetah's tail during a deceleration maneuver is analyzed based on motion capture data. In both these examples, forward rotation of the appendage in question is found to counteract the unwanted forward pitch caused by the braking forces. It is theorized that this stabilizing action may facilitate more rapid deceleration by allowing larger or longer-acting braking forces to be applied safely.

Tails, Flails, and Sails: How Appendages Improve Terrestrial Maneuverability by Improving Stability.

Collisions between rigid bodies are impulsive events, meaning they create near-instantaneous, finite discontinuities in the velocity state of the dynamic system. If velocity is assumed to be continuous, that requirement may come into conflict with the desire for bodies to not interpenetrate, or with Coulomb friction (creating a condition known as the Painlevé paradox). This letter presents a contact-implicit framework for trajectory optimization using direct collocation that combines an unplanned impulsive contact model with an implicit, high-order numerical integration scheme. We demonstrate that this framework can resolve problems that other implicit formulations cannot, namely elastic or partially elastic collisions, and impacts without collision – the established resolution of the Painlevé paradox. We then evaluate its applicability to legged robotics problems and test its ability to discover the intricate sequence of varied collision types necessary to execute a skateboarding trick.

Contact-Implicit Direct Collocation With a Discontinuous Velocity State

The linear-quadratic regulator (LQR) is an efficient control method for linear and linearized systems. Typically, LQR is implemented in minimal coordinates (also called generalized or "joint" coordinates). However, other coordinates are possible and recent research suggests that there may be numerical and control-theoretic advantages when using higher-dimensional non-minimal state parameterizations for dynamical systems. One such parameterization is maximal coordinates, in which each link in a multi-body system is parameterized by its full six degrees of freedom and joints between links are modeled with algebraic constraints. Such constraints can also represent closed kinematic loops or contact with the environment. This paper investigates the difference between minimal- and maximal-coordinate LQR control laws. A case study of applying LQR to a simple pendulum and simulations comparing the basins of attraction of minimal- and maximal-coordinate LQR controllers suggest that maximal-coordinate LQR achieves greater robustness and improved performance compared to minimal-coordinate LQR when applied to nonlinear systems.

Linear-Quadratic Optimal Control in Maximal Coordinates

With their combination of power and compliance, pneumatic actuators have great potential for enabling dynamic and agile behaviors in legged robots, but their complex dynam-ics impose control challenges that have hindered widespread use. In this paper, we describe the development of a tractable model and characterization procedure of an off-the-shelf double acting pneumatic cylinder controlled by on/off solenoid valves for use in trajectory optimization. With this we are able to generate motions which incorporate both the body and actuator dynamics of our robot Kemba: a novel quadrupedal robot prototype with a combination of electric and pneumatic actu-ators. We demonstrate both a 0.5 m jump and land maneuver, and a maximal 1 m jump, approximately 2.2 times its leg length, on the physical hardware with the proposed model and approach. The hardware matches the desired trajectory with a maximum height error of only 5 cm without any feedback on the pneumatic joints, demonstrating the utility of the model in high-level motion generation, and capability of the physical robot. 

Getting Air: Modelling and Control of a Hybrid Pneumatic-Electric Legged Robot

Modeling the Coulomb Friction Cone in trajectory optimization is typically done by linearizing it along a series of vectors. Often, these vectors define the edges of polyhedral estimations of the cone. This article provides an alternate approach that samples the cone along a vector that satisfies the Maximum Dissipation Principle, which is shown to be significantly more computationally tractable. The proposed technique uses the polar representation of the relative velocity of a contact point on a surface to determine the direction of the resultant friction force and linearizes the friction cone along this vector. This study describes the development of the proposed model. Thereafter, a trajectory optimization experiment was conducted to compare the traditional four-sided polyhedral estimation of the friction cone with the novel method. Compute time and computational complexity were used as performance metrics in this study. Results from these experiments indicate that the proposed method reduced the compute time by 39.13% and a 57.00% reduction in inequality constraints when compared to the 4-sided polyhedral estimation.

A novel method for computing the 3D friction cone using complimentary constraints

Many different coordinate formulations have been established to describe the position of multi-body robot models, but the impact of this choice on the tractability of trajectory optimization problems has yet to be investigated. Relative formulations, which reference the position of each link to its predecessor, reduce the number of variables and constraints in the problem, but lead to cumbersome expressions for the equations of motion. By contrast, referencing the positions to an absolute frame simplifies these equations, but necessitates more coordinate variables and connection constraints. In this letter, we investigate whether changing the orientation coordinates of a multi-body system model from relative to absolute angles can reduce the time required to solve the problem. The two approaches are tested on a variety of two- and three-dimensional models, with and without unscheduled unilateral contacts. Across all cases, the absolute formulation was found to be the more successful option. The performance improvements increased with the complexity of the system and task, culminating in the challenging example of a 60-degree turn on a 3D quadruped model, which was only able to converge in the allotted time when absolute angles were used.

Minor Change, Major Gains: The Effect of Orientation Formulation on Solving Time for Multi-Body Trajectory Optimization

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Minor Change, Major Gains: The Effect of Orientation Formulation on Solving Time for Multi-Body Trajectory Optimization