IROS2019国际学术会议论文集 0633VIP

Seeking the Analytical Approximation of the Stance Dynamics of the 3D Spring Loaded Inverted Pendulum Model By Using Perturbation Approach Haitao Yu1 Shengjun Wang1 Kaizheng Shan1 Jun Li1 Lixian Zhang2and Haibo Gao1 Abstract The Spring Loaded Inverted Pendulum SLIP has been widely exploited in both biomechanical and robotics research due to its simple form in mathematics and high accuracy in fi tting experimental biology data However the intrinsic nonlinearity of the SLIP dynamics makes accurate analytical representation unavailable Traditional methods take advantage of numerical integration to handle this issue while several existing analytical approximations focusing on 2D SLIP model The 3D SLIP suitable to physical reality is rarely investigated This paper presents a novel perturbation based approach to obtain the closed form analytical approximations of the 3D SLIP model in stance phase In contrast to existing work ignoring the gravitational forces the proposed approach just relies on assumptions of small leg compression and small leg swept angle The performance of the derived approximations has been evaluated via comprehensive numerical analysis The quality of accurate apex prediction promises the approximation as an advantageous and reliable tool for locomotion control of legged robots I INTRODUCTION Hopping running trotting and galloping are the most ubiquitous dynamical gaits displayed by cursorial mammals propelling themselves to traverse land surface Such agile energetic effi cient and robust behaviors when interacting with surrounding environment continually motivate engineers and robotics scientists to reproduce comparable performance on man made devices from monopode 1 biped 2 4 to quadruped robot 5 7 Since fi rst established in 8 the spring loaded inverted pendulum SLIP is utilized as a versatile template for both biomechanical studies 9 and legged robot development 10 11 due to its mathemat ical simplicity universal adaptability characterizing diverse legged locomotion with high fi delity of predicting the sagittal trajectory of the centre of mass CoM as well as the ground reaction force profi le 12 Benefi tting from these afore mentioned advantages a substantial SLIP centric theoretical This work was supported in part by by the National Natural Sci ence Foundation of China under Grant 51605115 Self planned Task No SKLRS201719A of State Key Laboratory of Robotics and Sys tems HIT Heilongjiang Postdoctoral Financial Assistance LBHZ16083 and Natural Science Foundation of Heilongjiang Province under Grant QC2017052 Corresponding author Haitao Yu 1Haitao Yu Sengjun Wang Kaizheng Shan Jun Li and Haibo Gao are with the State Key Laboratory of Robotics and Systems HarbinInstituteofTechnology 150080 Harbin Heilongjiang China yht shenjunwang kaizhengshan junli gaohaibo 2Lixian Zhang is with the Research Institute of Intelligent Control and Systems Harbin Institute of Technology Harbin 150080 China Lixianzhang results i e design methodology 13 14 control strategy 15 16 have come forth to provide potential possibilities for improving the dynamical stability and the performance of legged robots Despite conciseness in mathematical formulation the in herent strong nonlinearity of the SLIP model makes the ac curate analytical representation unaccessible due to the 2nd order non integrable terms contained in the stance dynamics 17 As an alternative pursuing an analytical approximation serving as the representation for the SLIP dynamics par ticularly in stance phase has become a valid way to cope with the model nonlinearity in recent years Early study on this fi eld starts with Schwind et al in 18 that a Picard type iteration working in conjunction with the mean value theorem is developed to derive a closed form approximation The approximate accuracy of the analytical results together with the algebraic complexity grows as the iteration steps increasing By neglecting the gravitational force Ghigliazza et al 19 propose a straight forward approximation of an ideal SLIP model wherein a fi xed leg reset policy is also presented to achieve stable hopping gait Based on the assumption of the angular momentum conservation Geyer et al in 20 derive a simple approximate solution for the planar SLIP model at stance capable of predicting symmetric trajectory of the CoM Arslan et al 21 enhance this solution by adding the gravity correction to compensate the angular momentum for the asymmetric case exhibiting a two step iteration format that ameliorates the prediction accuracy Shahbazi et al 22 extend this iterative procedure to obtain the approximation for the bipedal SLIP system at double stance phase providing an analytical return map to investigate the human like walking running and inbetween transitions In our previous work 23 the perturbation based analytical approximation for the stance dynamic of the planar SLIP model is derived with satisfactory accuracy of the apex prediction in both symmetric and asymmetric case In contrast to the well explored sagittal SLIP model with numerous theoretical fi ndings in both dynamical behaviors and locomotion control the Spatial SLIP SSLIP model a k a 3D SLIP a more generalized template for modeling walking and running behavior of animals and humans in 3D has not been adequately investigated and reported thus far By imposing the negligible gravity assumption equivalently high stiffness of leg Seipel et al decompose the movement of the 3D SLIP into the sagittal plane and the horizontal plane wherein the explicit approximate solutions for the re 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 IEEE3314 ductive integrable SLIP dynamics could be acquired deriving an analytical stride to stride Poincar e mapping digging into the dynamical behaviors over 3D space 24 Carver et al numerically investigate the discrete return map of the 3D SLIP dynamics with the touchdown angles and the piecewise constant leg stiffness as the control inputs presenting a stride to stride recovery scheme for small lateral perturba tions 25 Peuker et al extend the traditional fi xed leg touchdown policy into the lateral and the vertical adjustment resulting in the numerical computed based control strategies to achieve stable and robust running gait patterns in 3D 26 Up to now the analytical approximate solutions for the 3D SLIP model with satisfactory apex prediction performance have not appeared yet according to the existing literatures reported In this work we propose a novel perturbation based approach to deal with the strongly coupled nonlinearities in the spatial SLIP dynamics acquiring the straight forward closed form analytical approximate solutions that preserve high precision in predicting apex state To the best of the author s knowledge the reported solution is the fi rst ana lytical approximation with elementary functions that takes the gravitational force into consideration for the SLIP model spatially running in 3D The derived formulation doesn t rely on the assumption of the negligible gravity in stance phase and works valid in a considerable large combination of the associated model parameters The remainder of this paper is structured as follows The classical passive 3D SLIP model is briefl y reviewed in Sec tion II In Section III a novel perturbation approach is utilized to tackle with the non integrable nonlinear formulation of the 3D SLIP dynamics in stance phase deriving a closed form analytical approximation Section IV implements the performance evaluation of the derived approximation includ ing apex vector predicting for 3D SLIP running Finally this paper ends with conclusion and future work in Section V II THE 3D SLIP MODEL A The Passive 3D Spring Loaded Inverted Model The classic passive SLIP model executing 3D running is conventionally modeled as a point mass m attached to a massless leg equipped with a constant stiffness k as illustrated in Fig 1 a As an extension to planar SLIP model the full stride of a 3D SLIP model in running is composed of two sub phases the stance phase when the toe touches the ground and the fl ight phase when the leg loses contact The events between these two phases are defi ned as touchdown TD triggered from fl ight to stance and lift off LO triggered from stance to fl ight During fl ight the SLIP system could re position its massless leg for the coming TD Herein we uses two variables to specify the leg orientation w r t the inertial frame the swept angle measured from the positive z axis to the springy leg and the splay angle measured from the negative y axis to the leg projection on the x y plane as illustrated in Fig 1 a The complementary angle of at TD is defi ned as the angle of attack TD 2 TD while the apex is defi ned as the highest point in fl ight Fig 1 b shows a full stride from x y z o m k g l x y z Current Apex Stance Flight ascent a Bouncing Forward b x y z Flight descent Sequential Apex o Fig 1 The 3D SLIP model a the coordinates set up with illustration of the structural parameters The arrow marks the positive direction of the related angles b the apex apex stride for the 3D SLIP model where the apex vector at the i th step collects both the running speed and height with Si xai yai zai Tthroughout this paper Without loss of generality the 3D SLIP model is supposed to running towards the positive y axis as moving forward The leg length l retains l0 at rest in the fl ight phase which is further divided into the descent sub phase from the current apex to TD and the ascent sub phase from LO to the sequential apex In contrast to the classic SLIP system on the sagittal plane the additional leg splay angle is defi ned to characterize the heading direction of the 3D SLIP spatially running on the x y horizontal surface the current apex to the next during an entire gait We also assume that the toe on ground is modeled as a pivot where no slippage occurs in stance B Dynamics of the 3D SLIP Model In fl ight phase the movement of the CoM is solely governed by the gravity exhibiting a ballistic trajectory with x y 0 z g 1 where the Cartesian coordinates x y z Tcollects the spatial position of the CoM in the inertial frame The swing leg pre position policy the leg with the rest leg length l0maintains a constant angle of attack TDduring the descent period is employed for the SLIP system during the fl ight to prepare for the subsequent TD event 3315 In stance phase the movement of the CoM is governed by both the gravity and the rebound force of the leg spring Note that it is convenient to depict the CoM position in polar coordinates l Twith x lsin sin y lsin cos z lcos 2 Accordingly the Lagrangian for the 3D SLIP system in stance is given by L m 2 l2 l2 2 l2 2sin2 mglcos k 2 l l0 2 and the equations of motion for the stance phase can be directly derived as below l k m l l0 g cos l 2 l 2sin2 3 2 l l 2 cot 4 2 l l g l sin 2sin cos 5 Compared with the formulation of the passive 2D SLIP model hopping on the sagittal plane the stance dynamics in 3D contains more complicated non integrable coupled terms such that the accurate analytical solution is unattainable C Nondimensionalizing In order to reduce the parameter dependance we introduce a new time scale st with s2 g l0to generate the dimensionless variable l l l0 x x l0 y y l0 z z l0 k kl0 mg yielding Flight phase x00 y00 0 z00 1 6 Stance phase l00 k l 1 cos l 02 l 02sin2 7 00 2l 0 l 0 2 0 0cot 8 00 2 l0 l 0 1 l sin 02sin cos 9 where 0and 00represent the corresponding derivatives with respect to Hereafter we will treat the aforementioned non dimensional version instead of the original 3D SLIP dynamics in Section II B III ANALYTICAL APPROXIMATE SOLUTION BASED ON PERTURBATION APPROACH A Basic Assumptions Aiming to facilitate the implementation of the perturbation approach the following assumptions are used Assumption I Small radial compression 20 The maximum compression of the leg spring in stance is suffi ciently small so that we have1 0 0and z t tLO 0 Exceptions such as rebound stumble or falling down are regarded as failure cases that are eliminated out of the dataset The performance criteria to evaluate the predicting accuracy of the analytical approximations is given by the percentage error PE with PE xa 100 xa true xa appr xa true PE ya 100 ya true ya appr ya true PE za 100 za true za appr za true where the subscript true and appr denote the value com puted by the MATLAB numerical solver ode45 with the absolute and relative tolerances set at 10 6and the analytical approximations respectively B Main Results For the varying parameter family k TD ya za we divide each one into ten grids so that 105trials have been executed in simulation wherein only 76744 trials are feasible according to the aforementioned requirements of feasible running The simulation results are shown in Fig 2 and Fig 3 Fig 2 a shows the statistical results of the percentage errors over total feasible trails The mean values of the percentage errors for xa yaand zaare all below 6 which indicates that the proposed perturbation approach provides a satisfactory performance in predicting apex state for 3D SLIP running The maximum error comes to 10 03 with relatively low leg stiffness for predicting the hopping height za According to the simulation parameter settings that the leg swept angle starts within the range 5deg 25deg the resulting movements during the 3D SLIP running implies that the evolution of over the stance period will somewhat violate the small leg swept angle proposed in Section III A However the proposed perturbation solution still works well in dealing with relatively large leg swept angle maximum 60deg range approx The same phenomenon emerges in the condition with the non dimensional stiffness k varying within the range 10 50 These aforementioned evidences confi rm that the apex prediction performance of the proposed perturbation solution is not sensitive to the model parameter 0 2 4 6 8 10 12 Percentage Error Maximum ErrorMean Std 0 10 20 30 40 50 60 Percentage Error Maximum ErrorMean Std a Perturbation based Solution b Approximations in 36 Fig 2 Performance comparison between the perturbation based solution in this paper and the analytical approximations in 36 a The left column details the Mean Std value with 3 85 4 71 4 91 3 73 5 30 3 94 for xa ya za respectively The right column shows the Maximum errors with 9 30 7 94 10 03 for xa ya za respectively b The left column details the Mean Std value with 25 43 7 80 27 21 6 93 20 77 10 96 for xa ya za respectively The right column shows the Maximum errors with 43 86 48 79 57 64 for xa ya za respectively 101520253035404550 0 2 4 6 8 10 Percentage Error Non dimensional Stiffness Fig 3 Percentage error statistical diagram for xa ya zaas leg stiffness increasing from 10 to 50 variation and could maintain superior accuracy over a large combination of the associated parameters of the SLIP system Furthermore we repeat the same procedure to evaluate the predicting performance of the analytical approximation in 24 since this is so far the only existing approximate solution reported among the literatures The corresponding results are collected in Fig 2 b It is clear to see that the perturbation solution obviously outperforms the existing approximation 24 over the parameter family This ob servation could be explained as the perturbation solution is taken the gravitational forces into consideration during the solving process unfortunately which is ignored in the approximation derived by Seipel et al In general seeking the analytical approximation of the 3D SLIP dynamics in stance is a trade off between the mathematical simplicity and the predicting accuracy Although the proposed solution in this paper is more complicated in formulation but preserves higher accuracy in computing the apex return map with 3319 better performance on predicting apex state At last but not least the gravitational forces could not be ignored in stance phase when computing the passive dynamics of the 3D SLIP model In order to reveal how the leg stiffness infl uences the performance of the approximations we further plot the PE diagram with k increasing as shown in Fig 3 Obviously the PE values decreases dramatically with larger leg stiffness for xa yaand za This phenomenon can be explained the fact that larger stiffness gives rise to suffi ciently small compres sion during stance Conversely a soft leg brings relatively large compression which somewhat violates the assumption on small radial compression and therefore downgrades the performance of the proposed approximation Overall the proposed perturbation based solutions proposed in this paper could provide considerably high accuracy in predicting the apex state for 3D SLIP system running creating a favorable analytical representation that facilitates model based gait controller design without resort to numerical integration V CONCLUSION AND FUTURE WORK This paper presents a novel perturbation based approach for solving the nonlinear dynamics of a 3D running SLIP model in stance phase By imposing two reasonable assump tions on small radial compression together with small swept angle the regular perturbation approach has been employed to handle the strongly coupled nonlinearities of the 3D SLIP model The derived approximation as an analytical repre sentation of the system dynamics is composed of elemen tary functions in closed form providing accurate precision for apex state without resort to the traditional numerical integration to solve the SLIP dynamics The performance evaluation results indicate that the proposed perturbation solution performs better than the existing approximation over a relatively large parameter family with the percentage errors within 6 in predicating the apex state Additionally the prediction performance of the proposed solution is robust to the model parameter variation The future work will extend toward