Eigenmodes of Nonlinear Dynamics: Definition, Existence, and Embodiment into Legged Robots with Elastic Elements

Abstract

Pogo-stick bouncing or the spring loaded inverted pendulum represent fundamental dynamics models for hopping and running in legged locomotion. However, these conceptual models are in general of lower order than the elastic multibody dynamics of versatile segmented legs. The question how to embody these simple models into real robot leg designs still has not been completely answered so far. The concept of eigenmodes for linear systems provides a tool to separate high-dimensional, coupled dynamics in one-dimensional (1-D) invariant ones. However, the dynamics of segmented legs is in general nonlinear such that even the existence of periodic motions, as appearing typically in locomotion tasks, cannot be generally guaranteed without changing intrinsic dynamics behavior substantially by control. This paper extends the concept of eigenmodes, which is well-known for linear systems, to the nonlinear case. By proposing a method for selecting the design parameters of multibody systems such that desired eigenmodes are achieved, the problem of embodying fundamental locomotion modes into legged systems is resolved. Examples of practically realizable leg designs are provided, which proof the existence of invariant, 1-D oscillation modes in nonlinear, elastic robot dynamics. An experiment on a multilegged robotic system validates that energetic efficiency can be gained by the proposed approach. C(q 0 + wz, wż)wż ∂U (q) ∂q T q=q 0 +wz −M (q 0 + wz) −1 −M (q 0 + wz) −1q = wz T ⋆ q 0 +wz T ⋆ q 0

An Overview on Principles for Energy Efficient Robot Locomotion

“…Eigenmodes of linear dynamics have been widely used for rendering such motions based on the classic generalized eigenvalue problem (see works by Blickhan, 1989;Geyer et al, 2006;Kashiri et al, 2017c).To approach the performance and efficiency of the biological archetype, it is crucial to employ physical elasticity in drive units. To exploit the full dynamics of segmented legs, Lakatos et al (2017) described eigenmodes of non-linear dynamics. Such a complete modal model can enhance the system performance as the robot design targets match the segmented leg dynamics with a desired dynamics set based on template models, e.g., Spring-Loaded Inverted Pendulum (SLIP), and desired motion considerations.…”

Learning from outside the viability kernel: Why we should build robots that can fall with grace

“…This is a property of the system dynamics, and therefore the hardware design and not the controller design. Especially when building legged robots that try to mimic a SLIP-like behavior [27] [28], these aspects should be considered in addition to measures such as passive stability and energy efficiency.…”

“…This model can be fit very accurately to experimental data of many different running animals [17] [18], allows accurate prediction [19], and also has been used to design controllers for simulations [20][21] [22][23] as well as actual robots [24][25] [26]. Indeed, there has been a lot of effort to give legged robots SLIP-like behavior, either through mechanical design [27] [28] or control [24] [29]. The two dimensional view of the model, in the sagittal plane parallel to the direction of travel, represents a submanifold of the 3Dspace, and the results can be extended to 3D motion both in simulation [23] and hardware [30].…”

Dynamic Locomotion Gaits of a Compliantly Actuated Quadruped With SLIP-Like Articulated Legs Embodied in the Mechanical Design

“…2(b)) is expected to differ substantially. However, using the method of modal dynamics matching [23], kinematic, inertial and elastic parameters can be found such that they match. This is demonstrated for a mechanically straightforwardly implementable example.…”

“…However, all above mentioned robotic applications of versatile and dynamic legged locomotion require high bandwidth joint torque interfaces, which are technically costly, expensive and rather implausible from a biological point of view 3 . In our previous work [23], the theory of so-called eigenmodes of nonlinear dynamics and a corresponding embodiment procedure are proposed. This paper presents a compliantly actuated quadrupedal robot with articulated legs featuring a SLIP-like dynamical behavior embodied as eigenmodes in the mechanical system design.…”