Efficient Spatial Dynamics for Continuum Arms

Continuum robot dynamic models have previously involved a choice between high accuracy, numerically intensive models, and low accuracy, computationally efficient models. The objective of this paper is to provide an accurate dynamic model with low computational overhead. Our approach is to place point masses at the center of gravity of the continuum section, rather than along the robot’s backbone or centerline. This enables the model to match the robot’s energetic characteristics with many fewer point masses. We experimentally validate the model using a pneumatic muscle actuated continuum arm. We find that the proposed model successfully captures both the transient and steady state dynamics of the arm. INTRODUCTION Because of their tentacle-like dexterity and inherent compliance, continuum robots are well suited for tasks in cluttered, delicate, or unstructured environments. Inspired by muscular structures such as tongues [1], elephant trunks [2, 3] and octopus arms [4,5], continuum arms can elongate, contract, and bend at any point [6, 7]. In addition to compliance and dexterity, potential advantages over traditional rigid-link robots include reduced manipulator weight, better fault tolerance [8], and human ∗Address all correspondence to this author. (a) (b) (c) FIGURE 1: (a) Serial multisection continuum arm handling an object, (b) continuum sections as gentle fingers of a grasping manipulator (from [10]), (b) continuum sections as compliant limbs of a quadruped robot (from [11]) friendly interaction [9]. Based on these potential advantages, continuum robots have been increasing in popularity in recent years. However, the lack of accurate and computationally efficient dynamic models continues to prevent wider adoption of these robots, particularly in industrial settings where rapid motions are required. Continuum robots are made up of multiple sections, each of which can typically have its own curvature and bending plane DSCC2015-9932 1 Copyright © 2015 by ASME Proceedings of the ASME 2015 Dynamic Systems and Control Conference DSCC2015 October 28-30, 2015, Columbus, Ohio, USA Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 03/04/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use [12]. Each section can be thought of as a parallel manipulator with multiple variable length (and bendable) actuators. Due to various design choices, many (though not all) continuum sections bend into circular arcs [13,14]. In this paper we will consider one such design a pneumatically actuated robot that has been previously modeled both kinematically and dynamically in [15]. It uses Pneumatic Muscle Actuators (PMA) which are well-suited to use in continuum arms due to their high flexibility and power to weight ratio [16]. Serial multisection continuum arm prototypes (figure 1a) have been used for adaptive whole arm grasping [17], obstacle avoidance [18], navigation/inspection in obstructive environments [19,20]. Figure 1 shows some parallel multisection continuum robots. These can be used as grasping manipulators where each continuum section acts as a gentle finger (figure 1b shows handling of a glass beaker) and coordinated as limbs for locomotion (figure 1c). While the design of continuum robots has advanced rapidly in recent years, deriving dynamic models that are both accurate and efficient remains a formidable challenge. This is due to the complexity of both kinematic and mechanics models for these highly nonlinear devices. However, a number of useful approaches have been previously reported in the literature. Parametric models represent the continuous smooth bending of continuum arms [21] but the tradeoff is that they map the modeled robots configurations to a restricted set of motion ”shapes”. Theoretical modeling for an inextensible, unidimensional cable robot was reported in [22] but typical continuum arms have multiple degrees of freedom. Elliptic integrals were used in [23] to model continuum sections but it only accounted for statics of the robot without considering the gravitational potential energy. Therefore the model cannot be applied for dynamic simulations of macro-scale continuum arms where gravity effect is significant. Cosserat rod theory was used to model inextensible tendon actuated continuum sections in [24–26] but the authors did not attempt to address the computationally efficiency of the resulting model. The modal kinematics for continuum arms [27] combined the structural accuracy of curve parametric models and numerical efficiency/stability of modal methods. Utilizing the modal kinematics, spatial dynamics for continuum sections were proposed in [8, 15]. However, these models were based on integral formulations and therefore computationally intensive and not suitable for rapid simulations. Composed of a small number of rigid-linked segments or mass points along the length, lumped parameter models of continuum arms avoid complex integral expressions and produce efficient results. However, to approximate the smooth bending of continuum arms, many segments or mass points are required [28]. This significantly increases the overall degrees of freedom in contrast to actual number of controlled joint variables along with the computational complexity. The lumped model in [29] used curve parametric models for the derivation but has numerical instabilities for straight arm poses [8]. The model proposed in [30] only used 3 segments but does not represent the mass distribution of continuum sections. Further, all the lumped models so far focus solely on reducing computational complexity without analyzing the accuracy of the model in comparison to actual system in the energy domain. Employing the modal kinematics proposed in [12, 27], this paper presents a single mass, lumped parameter spatial dynamic model for continuum arms. Unlike previously reported lumped models, this approach investigates the mass-related energy of continuum arms to derive the best representation of the entire system using only a single lumped mass. The result is a model that is both simple and computationally efficient. Lastly, we experimentally validate the model using a prototype PMA actuated continuum arm. Note that while this paper was in the review process, the results described herein were extended to archival form and published in [31]. PROTOTYPE CONTINUUM ARM The prototype continuum section shown in figure 2a consists of three mechanically identical extending PMA’s [32] with unactuated length L0 = 0.22m, lower and upper bounds of the length variation lmin = 0m, and lmax = 0.071m within the operation pressure range (0−5bars). Due to the gradual pressure buildup in PMA’s, the maximum joint-space velocity is 0.16ms−1. Silicone tubes of inner diameter (ID) 10mm and outer diameter (OD) 13mm make the PMA bladders. Nylon union tube connectors (ID = 6mm) seal the Silicone tubes at either end. The pressure-supplying Polyethylene tubes connected to union connectors provide the pressure inputs. The Polyester braided sheath (OD = 13−26mm) is then inserted and tightened with high strength Nylon cable ties. Rigid plastic mount frames of r = 0.016m and 2.54mm thickness (see figure 2b, top right) are used to mount the PMAs. Rigid circular plastic constrainers (figure 2b, bottom right) help constrain PMA’s to operate parallel to the neutral axis at designated clearance ( 2π 3 rad apart at r ) from each other as well as provide improved torsional stiffness. The complete continuum arm has a mass (m) of 0.168kg. The pressure to each PMA is controlled by a Pneumax R © 171E2N.T.D.0009S digital proportional pressure regulator which is controlled through a RS232 digital command-response type interface that support real-time pressure commanding and reading. NDI Aurora tabletop magnetic tracking system provided the tip position at 30 times per second at a mean accuracy of 0.008m. Figure 3a shows the schematic of the prototype continuum arm shown in figure 2a. Variable length continuum arms generate motion by elastic deformation and this causes the length of the neutral axis to vary, hence the term variable length. As a result, the points along the arm have varying relative position and orientations. The length of an actuator at any time is L j = L0+ l j(t) where 0≤ l j (t)≤ lmax, j ∈ {1,2,3} and t is the time. 2 Copyright © 2015 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 03/04/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use