IROS2019国际学术会议论文集 0717

Kinematic Modeling of a Soft Pneumatic Actuator Using Cubic Hermite Splines Mats Wiese Kenneth R ustmann and Annika Raatz1 Abstract Soft material robotic systems provide increased adaptability and fl exibility compared to conventional rigid metal robots The soft systems benefi t from their inherent compliance which enables them to be used in applications that require safe interaction between humans and robots or manipulation in cluttered environment Despite advancements in recent years research on soft material robots still needs to make progress in terms of modeling for model based control or path planning The high nonlinearity of soft material robots makes effi cient and accurate modeling diffi cult In this work we introduce a kinematic modeling approach based on cubic hermite splines The method is applied to a soft pneumatic actu ator and evaluated against the widely used constant curvature approach The hermite spline offers the possibility of accurate shape reconstruction from simulated or measured deformation data Both the shape of a robot s segment and its orientation can be approximated this way In this paper a machine learning approach is used to train the kinematic relation between actuating pressure and confi guration parameters I INTRODUCTION While the kinematic modeling of rigid robots is a straight forward procedure today the kinematic modeling of soft robots is still characterized by certain challenges Rigid robots have discrete joints with discrete and a fi nite number of degrees of freedoms Soft continuum robots on the other hand exhibit full body deformation with theoretically an infi nite number of degree of freedom 1 This fact makes mod eling diffi cult because not every freedom can be modeled individually A lot of models have therefore been developed based on the fi nite element method in order to obtain a highly discretized representation of the continuum 2 3 Despite the level of detail and accuracy of these models their main drawback is the high computation time to solve a high dimensional system of partial differential equations Although there is ongoing research see 4 and 5 for example this prohibits FEM models from being widely used in control tasks or path planning To overcome this drawback other approaches for modeling include analytical models based on multi body systems 6 or continuum mechanics like the cosserat model 7 8 However analytical models are often hard to derive This obstructs the mass development of new and individualized soft robots One wide spread approach for modeling continuum robots deformation is the constant curvature approach also referred to as piecewise constant curvature PCC proposed by 14 In the past a multitude of soft robots have been designed in a way to be able to model their deformation using the constant All Authors are with the Institute of Assembly Technology Leibniz Uni versit at Hannover Germanywiese match uni hannover de curvature approach The approach represents the confi gura tion of a continuum robot segment as a curve with constant curvature in two or three dimensional space Although the approach usually neglects external forces like gravity it has been widely used in the soft robotics community as well 9 10 However in the presence of signifi cant external forces soft actuators exhibit nonconstant curvature Section V Moreover specifi c actuator designs like conically shaped segments also do not conform to the constant curvature assumption leading to a deviation between robot and model not only regarding the overall shape but also regarding the orientation along the robot s body The orientation of the actuator s tip is of signifi cance for specifi c operations or if multi segment soft robots are considered For shape reconstruction with only a few pa rameters slight errors in the orientation of a segment lead to a propagation of errors when the robot specifi c kinematic model is used for simulation of forward kinematics for example during path planning or kinematic synthesis We therefore focus on an alternative kinematic represen tation of soft robotic segments that deals with the drawbacks of the PCC approach in specifi c applications Note that we do not intend to replace the PCC approach but give an alternative for cases in which the approach reaches its limits To overcome these limits variable curvature approaches have been proposed in the past In 11 a variable curvature approach is presented that further divides an actuator segment into further segments with constant but different curvatures However this goes along with a higher dimensional confi g uration state space Before that 12 13 proposed a modal approach for hyper redundant robots and continuum robots that relies on a set of modal shape functions The kinematic modeling approach presented in this work is part of a previously developed design and modeling frame work for soft continuum robots 15 17 The framework is built upon the capability of fi nite element analysis FEA a simplifying kinematic modelling of the deformation and machine learning of the robot specifi c kinematic relationship between actuating pressure and confi guration To date the kinematic modeling of a robot segment s deformation relies on the PCC approach The work presented in this paper aims for a more fl exible kinematic modeling approach that can be applied to nonconstant curvature deformation It provides an accurate yet low dimensional way to parametrize the shape of an infi nite dimensional soft continuum robot The remainder of this article is organized as follows In Section II the embedding framework is briefl y introduced Subsequently the kinematic modeling approach using cubic 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 IEEE7176 hermite splines is presented We will deal with the recon struction of the shape as well as with the orientation along the so called backbone curve The spline approach is applied to a soft pneumatic 3D bending actuator in Section IV Therein the approach is evaluated against the PCC approach using simulated deformation data with different loading scenarios including gravitational forces In Section V a spline is fi tted to experimentally obtained deformation data to prove the feasibility of the approach on a real 3D bending actuator Section VI incorporates the spline approach in the previ ously developed framework The robot specifi c relationship between actuation input and confi guration output under the infl uence of gravity is learned for both a PCC and a spline representation and evaluated against each other Finally Section VII summarizes the main contributions of this article and gives an outlook on future research II SOFTROBOTICSDESIGN ANDMODELING FRAMEWORK In recent works we presented a methodological frame work for modeling soft pneumatic actuators The framework supports a largely automated kinematic modeling of a soft pneumatic actuator starting with a generic design up to the machine learning of a kinematic relationship Therefore the modeling process is divided into four steps In the fi rst step a soft pneumatic actuator model is de signed by defi ning its geometrical parameters and material properties Utilizing these parameters a fi nite element model is derived in a second step and simulated under multiple pressure loads With this procedure a robot specifi c relation ship between pressure inputs and deformation is determined which is often diffi cult to derive analytically Deformation data is in a third step used to compute the robot specifi c for ward kinematics To date this step utilizes the PCC approach and an extension of the conventional approach presented in 15 Therefore the deformation data obtained by FEA is condensed to a single backbone curve that describes the macroscopic deformation of the soft robotic segment This backbone curve is then approximated by fi tting a circular arc to the data in order to derive the confi guration parameters The PCC approach also allows for a robot independent map ping between the confi guration parameters and end effector position of a soft robotic segment This is achieved by parameterizing a homogeneous transfer matrix using the con fi guration parameters This article deals with an alternative representation of the robot s confi guration based on cubic hermite splines In a fi nal step an artifi cial neural network ANN is trained for the relationship between pressure and confi guration using the data set obtained from the second and third step While the derivation of the robot specifi c relationship based on FEA requires extensive computation a once trained ANN is a computationally effi cient model that represents the robot specifi c kinematic relationship Since simulated data is used for the training the time consuming and costly generation of training data on the real actuator is no longer necessary during the design stage III KINEMATIC MODELING USING CUBIC HERMITE SPLINES In order to be able to represent the deformation of a soft robot segment the deformation along three equally spaced paths is imported into MATLAB A path is therefore defi ned by n points on the actuator s outer surface along its length Here pk irepresents the i th point of the k th path starting from i 1 at the base of the actuator to i n at the tip of the actuator see Fig 1 A backbone curve can be computed with help of these coordinates pi p1 i p2 i p3 i 3 1 While in most publications the backbone curve of a soft robotic segment is approximated with a constant curvature approach this is a simplifi ng assumption that does not hold for every soft robotic design or under every condition Especially external loads like gravity can lead to a deviation from constant curvature deformation The use of curves with variable curvature offers a more fl exible approach We therefore propose the use of a cubic hermite spline The remainder of this section deals with the mathematical description of the approach namely the approximation of the backbone curve in III A and the derivation of a homogenous transformation in III B A Backbone curve approximation using cubic hermite splines A cubic hermite spline is a hermite function valid on a certain domain interval Given its end points and the corre sponding derivatives orientation the cubic hermite spline is defi ned This predestines these splines for use in continuum robotics where position and orientation are important A cubic hermite spline is based on four cubic polynomial basis functions h0 s h1 s h2 s and h3 s 18 x s h0 s x0 h1 s x1 h2 s v0 h3 s v1 2 with h0 s 1 3s2 2s3h1 s 3s2 2s3 h2 s s 2s2 s3h3 s s2 s3 3 p1 i p2 i p3 i pi pi 1 Fig 1 Visualization of the backbone curve computation utilizing three discretized paths 7177 The spline is defi ned by two points and two vectors In case of a continuous robot the defi ning points are the base and tip position x0and x1 respectively The two vectors v0and v1 represent the tangent direction at the base and the tip of the actuator thus they are a represantation of the orientation The lengths of these vectors determine how much they infl uence the curves behaviour In order to fi t a cubic hermite spline to a given backbone curve the following conditions are formulated 1 The base position of the spline matches the base position of the backbone curve which is at the origin x0 0 00 T 4 2 The tip position of the spline matches the tip position of the backbone curve x1 pn 5 3 Assuming the actuator is clamped at its base the tangent vector points in z direction v0 m0 0 01 T 6 4 The tangent vector at the spline s tip points in the same direction as the backbone curve whose direction is given by the vector that is perpendicular to the plane spanned by the end points of the three sampled paths v1 m1 p1 n p2 n p1 n p3 n k p1 n p2 n p1 n p3 n k 7 The scaling factors m0and m1of the tangents leave room for an optimization to not only fi t the position and orientation at the base and the tip but to also approximate the overall shape of the backbone curve To do so the mean distance d m between backbone curve and the spline is computed by sampling x s and mapping the individual points of the backbone onto the linearly interpolated segments between the samples To fi t a cubic hermite spline to the given backbone curve this distance is minimized with respect to the scaling factors m0an m1 m argmin m d m 8 To solve the minimization problem the Levenberg Marquardt solver for nonlinear functions provided by MATLAB is used B Homogenous transformation based on cubic hermite splines The cubic hermite spline itself only represents the shape of an actuator s backbone curve In addition to that the orientation along the curve is of interest especially when it comes to multisection soft robots or the attachement of an end effector whose orientation is of importance A homogenous transformation is given as follows TFS s R s x s 0001 9 where R s n s b s t s 10 and t s x0 s x0 s 11 0 denotes the derivative with respect to s t s is the unit tangent vector to the curve n s and b s are called normal unit vector and binormal unit vector respectively These vectors form an orthonormal basis at the point x s of the curve A natural way to frame a curve and determine n s and b s is the Frenet Serret formalism 12 However for multisection continuum robots the Frenet Serret frames are not suffi cient to express the orientation at a point x s along the backbone curve of the actuator While according to the Frenet Serret formalism the normal vector always points to the center of the respective osculating circle the actual orientation of the actuator is different 12 Assuming minimal torsion of the actuator s backbone curve the double refl ection method as proposed in 19 is applied to solve the orientation problem The frame attached to the curve at the base of an actuator segment corresponds to the global coordinate system meaning n0 1 00 T b0 0 10 Tand t0 0 01 T The tangent at every point s on the curve is calculated with 11 ti t si 12 The fi rst refl ection involves the mapping of the initial right handed frame R0to a left handed frame RL 0 by utilizing the refl ection across the bisecting plane between x0and xi The second refl ection maps the left handed frame to a right handed frame Ri using the refl ection across the bisecting plane between x1 tL 0 and x1 t1 Fig 2 visualizes the procedure On the left hand side the fi rst map is depicted the right hand side shows the second map For effi cient computation of the frame only the unit vectors n and t are mapped b is computed with 13 bi ti ni 13 In contrast to the PCC approach the spline approach with the proposed determination of orientation can cope with torsional deformation of the actuator The deformation representation of a soft robot obtained with a cubic hermite spline is defi ned by eight parameters namely the three dimensional position of the curve s end x1 the three dimensional unit tangent vector v1and the two dimensional vector of scaling factors m IV MODELING OF A3DBENDING ACTUATOR The kinematic modeling approach using cubic hermite splines is applied to a 3D bending actuator described in previous publications 15 17 and depicted in Fig 3 The actuator consists of Ecofl ex 50 and Dragon Skin Three pressure chambers are equidistantly distributed around the circumference and along the actuator length The actuator can bend in three dimensional space depending on the applied pressure in the pressure chambers To prevent the actuator s tip from deforming extensively a 10mm thick 7178 tL 0 nL 0 n0 t0 b0 tL 0 nL 0 ti ni bi x0 xi Fig 2 Double Refl ection method for computation of rotation minimizing frames On the left hand side the fi rst map is visualized the right hand side shows the second refl ection 19 Fig 3 Exterior view and sectional view of a soft pneumatic actuator layer of the stiffer silicone Dragon Skin is applied on top of the actuator This ensures a more symmetrical deformation of the actuator segment In the following two cases are examined in simulation First the actuator segment is actuated with a pressure of 230mbar in one chamber This is done without consideration of gravitational forces Secondly 232mbar is applied to the fi rst two chambers while gravity acts in radial direction to the undeformed actuator For both cases the mean deviation between the simulated reference backbone curve and confi g uration is determined for the spline approach and for the con ventional and extended PCC approach The three approaches are visualized in Fig 4 For further information refer to 14 for the conventional and 15 for the extended PCC approach Moreover the orientation error at the actuator s tip is determined by calculating the angular error between the tangent normal and binormal direction of the respective frames The tangent of the backbone curve is computed similar to 7 with help of the backbone curve samples tbbc p1 n p2 n p1 n p3 n k p1 n p2 n p1 n p3 n k 14 The normal direction is calculated with help of one of the defi ned paths which in the undeformed state is defi ned to lay opposite the fi rst pressure chamber on the x axis and parallel to the z axis In the deformed state this corresponds to the normal direction of the frame that is attached to the Fig 4 From left to right constant curvature arc constant curvature arc with linear segments near the curve s ends extended PCC cubic hermite spline backbone curve nbbc pn p1 n kpn p1 nk 15 The binormal direction is calculated analougosly to 13 bbbc tbbc nbbc 16 By defi nition the frame at the actuator s tip is the same for the conventional and the extended PCC approach A Modeling Without Consideration of Gravitational Forces Fig 5 shows the backbone curve that results from an actuation with 230mbar in one chamber and without consid eration of gravity In comparison the approximating confi gu rations obtained by the three different approaches are shown For a detailed explanation on how