IROS2019國際學術會議論文集 1287

Confi guration Modeling of a Soft Robotic Element with Selectable Bending Axes Emily A Allen1 Brandon C Townsend1and John P Swensen1 Abstract This paper presents an approach for modeling new soft robotic materials which possess the ability to control directional stiffness These materials are inspired by biological systems where movements are enabled by variable stiffness tissue and contraction of localized muscle groups Here a low melting point LMP material lattice embedded in an elastomer serves as a rigid skeleton that may be locally melted to allow bending at selectable joint locations The forward kinematics of the lattice has been modeled using the product of exponentials method with the incorporation of bending axis selectivity In this paper we develop this model to account for torques imposed by tendons and we model the elastomer s resistance to bending as a torsional spring at the selected joints Thus we obtain a two way relationship between tendon forces and joint angles axes The concept of applying traditional robot modeling strategies to selectively compliant robotic structures could enable precise control of dexterous soft robots that satisfy stringent safety criteria I INTRODUCTION Several novel approaches to soft robotics actuation have been innovated in the last several years as researchers explore the soft robotics alternative to traditional rigid robots With the growing demand for at home healthcare and the push for industrial co robots safety and adaptability are a high priority for modern robots 1 The intrinsically soft nature of the soft robotics approach offers solutions to safety concerns and shows great promise for mimicking human abilities 2 4 Unlike traditional robots soft robots can deform upon impact to prevent injuries Several approaches have been developed to address the strength fl exibility trade off introduced by the soft robotics approach 1 5 In an effort to maintain the structural integrity of tradi tional robots variable stiffness actuators operate by varying the stiffness transmitted to joints between rigid links An tagonistic arrangements of actuators mimic human muscle confi guration and can exhibit a nonlinear relationship be tween input torque and angular joint defl ection 6 8 Other methods use advanced control systems to enable variable stiffness Although these methods offer high level precision and reliability their practicality is limited by size weight and bandwidth and they are generally not suitable for material actuators with multiple degrees of freedom 9 10 Variable stiffness structures are comprised of prestressed struts and cables that hold the structure in a confi guration by This research was funded by the National Science Foundations National Robotics Initiative Award 1734117 1Emily Allen BrandonTownsendandJohnSwensenarewith theSchoolofMechanicalandMaterialsEngineering Washington State University Pullman WA 99164emily allen2 wsu edu brandon townsend wsu edu john swensen wsu edu selectively releasing and re tensioning cables within a robotic system a variety of predictable motions may be activated 11 13 Similarly the application of 4D printing to soft robotics has enabled assemblies that self bend when exposed to light heat electricity or other means of stimulation A patterned innk deposition causes heating shrinkage at folding sites or localized swelling of media within polymer matrices under infrared exposure 14 15 Other researchers have designed robotic structures using origami techniques 16 17 Soft robotic actuators for specifi c applications are often designed to meet the compliance requirements of the system at hand 18 Many soft actuators are fabricated with geometrically patterned pneumatic chambers that deform the elastomer when pressurized 19 23 Although elastomers are inherently weak fi ber reinforcement and high pressure supplies offer high strength capabilities 4 24 25 Research similar to the work in this paper is driven by a need for soft robotic structures materials that can exhibit both high strength and compliant behavior The impressive capa bilities of biological systems such as muscular hydrostats and catch connective tissue inspire the design of soft robotic materials whose stiffness can be precisely controlled 26 27 Some researchers have explored the use of low melting point LMP materials to enable stiffness variability 28 33 Heating of these materials causes the internal skeleton to melt and allow compliant behavior when desired These concepts are expanded to enable directional stiffness control by locally melting the skeleton at designated loca tions 34 In this work we propose a method for modeling the confi guration of an element with selectable bending axes Fig 1 Design of 3 link soft robotic element with internal LMP lattice that can be selectively melted to allow bending about 9 different axes The colored arrows represent the 9 selectable bending axes denoted by i j where i refers to the segment number and j is the axis direction The points q1 q2 and q3lie at the centers of the 3 segments 2019 IEEE RSJ International Conference on Intelligent Robots and Systems IROS Macau China November 4 8 2019 978 1 7281 4003 2 19 31 00 2019 IEEE4353 A 3 link soft robotic element has been proposed with an internal rigid skeleton that may be selectively melted to allow bending about specifi ed axes as shown in Fig 1 The element consists of a low melting point LMP metal or polymer lattice encased in silicone rubber with nichrome heating elements arranged to allow selection of bending axes Fig 2 Planar view of tendon routing for 3 link soft robotic element when the 3 transverse axes are melted simultaneously The nine selectable bending axes represented by i j are shown in Fig 1 where i refers to the segment number and j is the axis direction For this problem up to 3 of the 9 bending axes may be selected at once up to one axis per segment by localized melting of the lattice As shown in Fig 2 a tendon is attached to each side of the element to induce bending about the selected compliant axes The forward kinematics of this element have been constructed using the product of exponentials method to determine the confi guration based on the selected axes and corresponding joint angles 34 A variety of unique confi gurations may be achieved as shown in Fig 3 with minimal complexity Fig 3 A few unique confi gurations achievable through selective melting and deformation of soft robotic element For this paper we take this model a step further by relating applied tendon forces to the deformation of the piece This involves modeling the joints melted axes as torsional springs with some constant stiffness By relating tendon forces to the elements confi guration we may determine what confi gurations are possible for any given axis selection Each set of equations is derived as a function of the set of selected axes which may include a single axis or up to 3 of the 9 possible bending axes Each time that a segment is deformed and then cooled the reference confi guration of the model changes and must be updated By successively melting different joints and controlling tendon forces a vast range of confi gurations may be achieved by this simple element Selective melting drastically improves the work space of the device allowing for fi ner control using only a single tendon When individual joints may be selected to melt on their own or even in pairs rather than all three the shape of the device may be more precisely controlled Simultaneous melting only allows for the bending of all selected joints at once with each joint experiencing approximately the same angular displacement This provides both little control and limited workspace Selective melting allows for confi gura tions and tool tip positions that would not otherwise be achievable by allowing for the manipulation of individual joints while the other joints remain fi xed II MATERIALS AND METHODS A Extension Functions for Tendons The torque applied on the joints by the tendons depends on the tendon routing confi guration and the axes that have been selected For example if the tendon does not lie perpendicular to the selected bending axis a larger tendon force will be required to achieve the same torque about the joint These geometric relationships between tendon force and joint torque are derived as a function of each possible axis selection joint angle and tendon offset a The joint torques may be directly related to the tendon forces by developing extension functions for each tendon This method of analyzing inelastic tendons is described by Murray et al 35 This method involves deriving the extension function for each tendon which expresses the length of the tendon as a function of the joint angles In our case since the axis directions may vary the extension functions depend on both the joint angles and the selected axes For a simple planar problem developing these extension functions may be done by simply analyzing the geometry For example if the axes 1 3 2 3 and 3 3 transverse axes are selected the geometric relationships may be extracted by inspection of the planar diagram in Fig 2 However when different axes are selected the problem is no longer planar and these geometric relations become nontrivial Rather than developing complicated three dimensional geometric relationships for each bending axis combination the forward kinematics exponential may be used to express the length of each tendon for any set of selected axes and joint angles The length of tendon 1 as shown in Fig 2 is simply the sum of the distances between tendon fi xation points 1 and 2 2 and 3 3 and 4 These distances are already known from the forward kinematics for this element which have been previously developed 34 x1 1 1 1 0 0a1 1 x2 1 1 1 e b 1 1 1g1 2 0 0 0a1 2 where 1 1 is the twist used to represent the rotation and translation of points due to bending about the selected axis 1 Here the second subscript indicates the frame of 4354 reference so x2 1represents the homogeneous coordinates of point 2 relative to frame 1 The matrix exponential e 1 1 used to transform points from frame 1 to 2 has been previously developed as a function of in the construction of the forward kinematics relationships for this element 34 The reference confi guration g1 20is the transformation between frames 1 and 2 when 1 0 For this case g1 20 100 1 0100 0010 0001 3 where 1is the length of link 1 The distance between x1 1 and x2 1may then be computed as d1 2 1 1 q x 1 1x1 1 2 x 2 1x2 1 2 4 Finally when the joint angles are all positive the extension function for tendon 1 may then be computed by summing the distances between each tendon fi xation point h1 d1 2 d2 3 d3 4 5 where d2 3and d3 4are computed using the matrix exponential for rotation about 2and 3 The extension function for tendon 2 may simply be expressed as h2 1 a 1 2 a 2 3 a 3 6 when the joint angles are all positive In theory there are 8 different cases for these extension functions based on different combinations of positive and negative joint angles For example if 1and 2are positive while 3is negative the extension functions would behave differently than if all the joint angles were positive Thus extension functions are different for each of the 8 cases of positive negative joint angle combinations However for this project we only have two tendons so if we consider only simultaneous melting the only possible joint angle combinations are case 1 all joint angles are positive and case 8 all joint angles are negative For case 8 the joint angles are all negative i e tendon 2 is activated instead of tendon 1 and the extension functions are reversed as follows h1 1 a 1 2 a 2 3 a 3 7 h2 d5 6 d6 7 d7 8 8 B Coupling Matrix According to Murray et al 35 by applying the conser vation of energy the joint torques can be expressed as P f P f 1 f2 9 where f is a vector containing the forces on each tendon and where P is the coupling matrix computed from P h 11 Here h is a vector containing the extension functions for the appropriate case of positive negative joint angle combinations h h 1 h2 12 The computation of P is non trivial Since the extension functions h depend on the joint angles and selected axes and they involve matrix exponentials and square roots taking the derivatives for P by hand would be tedious and nearly impossible The extension functions for all 8 cases were entered into Mathematica for the analytical computation of these derivatives The resulting coupling matrix for each case expressed as a function of a 1 2 and 3 was then converted to MATLAB using the ToMatlab package C Measuring Joint Stiffnesses The joint stiffness modeled by the torsional springs is dependent on the geometry of the element at the joint and the material properties of the elastomer By assuming Hooke s Law behavior the torque exerted by a single spring joint can be expressed as k 13 where is the resulting joint angle relative to the equi librium confi guration and k is the spring constant which may be determined experimentally k depends on the elastic modulus of the material and the area moment of inertia of the joint cross section For this particular project there are only two different cross sections for the nine different allowable bending axes a cross section for the straight transverse axes and a slightly wider cross section for the diagonal axes A primitive experiment setup shown in Fig 4 was used to measure the effective spring constants of these melted joints on the soft robotic element The desired axis was heated to 60 C to melt the polycaprolactone PCL polymer lattice along the axis then the tendon was pulled with a load cell to measure the perpendicular force required to incur a 90 bend at the axis average force measurements from 3 identical repeated tests were then converted to torsional spring constants listed in the table below These values provide primitive estimates of the joint bending stiffnesses but further testing is needed to confi rm the constant stiffness assumption The appropri ate stiffness value kstraight kdiagonal or ksolid is selected in MATLAB based on the input axis selections Fig 4 Setup for experimental testing of effective torsional spring constants for a transverse bending axis and b c diagonal bending axis 4355 TABLE I MEASUREDTORSIONALSPRINGCONSTANTS FORSTRAIGHT AND DIAGONALAXES MeasuredVariable Axis TypeSpring ConstantName Straight melted 21 7 1 4 N mm radkstraight Diagonal melted 27 0 1 0 N mm radkdiagonal Straight solid 115 11 N mm radksolid D Relating Potential Energy to Joint Torques The Euler Lagrange method may be used to develop the equations of motion for the robotic element Since we are only dealing with the statics of this problem the higher order terms may be neglected In other words we can neglect the effects of kinetic energy and rotational inertia on the system We will also choose to neglect gravity for this paper By simplifying the problem in this way the resulting governing equation will take the form L K 14 where L is the Lagrangian is the joint angles and is the joint torques vector Given that only potential energy is being considered in the system the Lagrangian is simply equal to the negative of the potential energy from the joints The torques generated by the bending at the joints are accounted for in the potential energy terms PE L 1 2k1 2 1 1 2k2 2 2 1 2k3 2 3 15 with kibeing the stiffness of the joint and ibeing the angular displacement at that joint Differentiating the Lagrangian with respect to yields L k 1 1 k2 2 k3 3 17 Applying the Euler Lagrange equation to the simplifi ed static model yields the following relationship L k 1 1 k2 2k3 3 18 By applying the relationship from 9 the joint torques may be expressed in terms of the tendon forces and coupling matrix Combining 9 and 14 produces the relationship K P f 19 where K is the joint stiffness matrix K k1 00 0k2 0 00k3 20 In our system we want to be able to determine given a particular f and We also hope to determine f given and To determine given f and we left multiply each side of 19 by the inverse of the stiffness matrix to yield K 1 P f 21 The vector of joint angles cannot be solved analytically in this expression since the coupling matrix P is a compli cated function of and since appears on both sides of 21 This equation can be solved numerically using fi xed point iteration to determine the joint angles that result from a given set of selected axes and tendon forces f To fi nd the forces required to produce a desired set of joint angles we simply left multiply 19 by P inverse to obtain f P 1 K 22 III RESULTS AND DISCUSSION A Confi guration Computation A relationship has been developed that enables computa tion of the joint angles resulting from application of a given tendon force For example Fig 5 a shows the computed equilibrium confi guration when a 4 Newton force is applied to the upper tendon and the heating elements are activated to allow bending along axes 1 1and 3 3 Other unique confi gurations may be achieved by applying different tendon forces and selectively melting different axes along the lattice as seen in Fig 5 b and c Fig 5 Simulated confi gurations of 3 link element under different heating and tendon loads where red lines indicate activated heating elements Various confi gurations may be used to perform intelligent tasks For example applying a 14 Newton force to the upper tendon while melting parallel diagonal axes could allow the robot to grab a pen as shown in Fig 6 Fig 6 Melting parallel diagonal axes allows element to wrap around pen B Workspace Limitations It is interesting to note that 21 fails to compute realistic joint angles if a joint angle exceeds 90 along a diagonal axis At fi rst glance this may appear to be a computational error but in reality this computational limitation perfectly 4356 represents a physical limitation Careful inspection of the tendon routing shown in Figs 1 and 2 reveals the reason for this limitation For this par