Hindawi Applied Bionics and Biomechanics Volume 2020, Article ID 8854411, 15 pages https://doi.org/10.1155/2020/8854411 Research Article Leg Locomotion Adaption for Quadruped Robots with Ground Compliance Estimation Songyuan Zhang , Hongji Zhang, and Yili Fu State Key Laboratory of Robotics and System, Harbin Institute of Technology, 150001, China Correspondence should be addressed to Songyuan Zhang; zhangsy@hit.edu.cn Received 18 July 2020; Revised 25 August 2020; Accepted 8 September 2020; Published 22 September 2020 Academic Editor: Liwei Shi Copyright © 2020 Songyuan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Locomotion control for quadruped robots is commonly applied on rigid terrains with modelled contact dynamics. However, the robot traversing diﬀerent terrains is more important for real application. In this paper, a single-leg prototype and a test platform are built. The Cartesian coordinates of the foot-end are obtained through trajectory planning, and then, the virtual polar coordinates in the impedance control are obtained through geometric transformation. The deviation from the planned and actual virtual polar coordinates and the expected force recognized by the ground compliance identiﬁcation system are sent to the impedance controller for diﬀerent compliances. At last, several experiments are carried out for evaluating the performance including the ground compliance identiﬁcation, the foot-end trajectory control, and the comparison between pure position control and impedance control. 1. Introduction proprioceptive design allows the force proprioception which can deliver the desired force with motor current sensing [10]. Currently, the main forms of locomotion robots include leg- For quadruped robot to achieve better performance, foot ged robots and tracked/wheeled robots. Compared with a trajectory planning is crucial. Two factors should be consid- tracked or wheeled robot, legged robots can easily adapt to ered in the design of the foot trajectory. One is that the great and walk on rough terrain [1]. Legged robots can choose con- impact should be avoided when the feet land on the ground; tact points with environment for overcoming obstacles and another is that the leg structure should have enough ground ﬁnding the feasible stable region [2]. For example, Boston clearance for avoiding obstacles [11]. The foot trajectory also requires continuous velocity and acceleration for stability. Dynamics developed a hydraulic driven BigDog robot, which aimed at building an unmanned legged robot with rough- There are three main methods to plan the trajectory of a terrain mobility [3]. MIT Cheetah was designed with propri- quadruped robot, i.e., Bézier curve trajectory, cubic trajec- oceptive actuator for impact mitigation and high-bandwidth tory, and sinusoidal trajectory [11]. Among them, the Bézier physical interaction [4]. In our previous research, inspired by curve meets the above requirements perfectly. After the foot the proprioceptive actuator [5], we designed a single-leg plat- trajectory planning, the behaviour that the robot interacts form for high-speed locomotion. ANYmal used the compli- with irregular ground should be considered. Quadruped cated actuator which makes the robot impact with high robots which can adjust the foot-end dynamic behaviour to torque control accuracy [6]. Overall, the hydraulic actuator deal with the unstructured terrains are important on real has the feature of naturally robust against impulsive loads application. For example, Semini et al. implemented an with a high-power density [7, 8]. However, legged robots impedance controller to control the electrohydraulically with hydraulic actuators such as Bigdog developed by Boston driven leg of HyQ [12]. Hyun et al. realized the virtual leg Dynamics are diﬃcult to be scaled down and will generate compliance of the MIT Cheetah with proprioceptive imped- large noise [9]. Diﬀerent from that, the electric actuator with ance control to deal with the external disturbance. After 2 Applied Bionics and Biomechanics Modeling · · e ,e ,e ,e F ,F 𝜏 ,𝜏 𝜌 𝜌 𝜃 𝜃 Geometric Impedance 1 2 1 2 Robot system transformation controller 𝜌 ,𝜌 ,𝜃 ,𝜃 d d d d · · p ,p ,p ,p x x y y F ,X ,X Ground system G f f State estimator identification Trajectory Planner · · · 𝜌, 𝜌 ,𝜃,𝜃 q ,q ,q ,q 1 1 2 2 Geometric transformation Figure 1: Overall of the control diagram. BHipR B Z BHipP BH BH BHipP BHipR X X BH BHipR BHipP Knee Knee Ankle Ankle Ankle Foot Foot Foot Figure 2: Link coordinate systems for the right front leg of the quadruped robot. knowing the coordinates in the Cartesian coordinate system, the estimated stiﬀness of ground will be used for adjusting the joint coordinates are obtained by inverse kinematic trans- the parameters of impedances controller. formation, and the joint motion trajectory is obtained. More- This paper is structured as follows: Section 2 intro- duces the kinematic modelling method for the leg. In Sec- over, the whole leg movement of quadruped robots will be divided into stance phase with impedance control and swing tion 2, the foot trajectory of the leg is designed utilizing phase with position control [13]. However, rigid ground the Bézier curve and sinusoidal waves. The detailed imple- assumption is applied for most researches. The diﬀerent con- mentation of impedance control and ground compliance tact dynamics between the rigid assumption and soft contact identiﬁcation will be introduced in Section 2. The experi- mental results are provided in Section 3. At last, the con- will aﬀect the performance and stability of robots. In contrast to that, Kim et al. penalizing the contact interaction in the clusions are given. cost function during the design of the whole-body controller [14]. Neunert et al. used the soft contact model combined 2. Materials and Methods with MPC controller [15]. Bosworth et al. designed a control- ler which can be tuned for diﬀerent ground types [16]. Diﬀer- The detailed leg design with three-joint leg structure can be ent from that, in our study, the least-squares method is found in our previous research [5]. The overall of the control applied to estimate the ground compliance parameters. The diagram is shown in Figure 1. During the movement of method is a soft terrain adaption algorithm which can robots, the Cartesian coordinates of the foot-end are achieve a transition between hard and soft ground with a obtained through trajectory planning, and then, the virtual real-time terrain-aware. For real leg locomotion adaption, polar coordinates in the impedance control are obtained Applied Bionics and Biomechanics 3 2 3 Z 10 0 0 HipR BH 𝜃 1 Z HipR Z HipP 6 7 L 6 7 Z 0 cosðÞ −θ −sinðÞ −θ 0 Knee 1 1 6 7 L 3 BH Y T = : ð1Þ 6 7 BH HipR HipP 6 7 HipP 0 sinðÞ −θ cosðÞ −θ 0 Y 1 1 X X 4 5 BH HipR Knee Knee 4 00 0 1 Ankle Ankle X B Ankle The homogeneous transformation matrix of the hip pitch coordinate system relative to the hip rolling coordinate Figure 3: One leg kinematic model of the quadruped robot. system is 2 3 cosðÞ θ 0 sinðÞ θ 0 2 2 6 7 through geometric transformation. The deviation from the 6 7 01 0 −L HipR 6 7 planned and actual virtual polar coordinates and the T = : ð2Þ 6 7 HipP 6 7 expected force recognized by the ground identiﬁcation sys- −sinðÞ θ 0 cosðÞ θ 0 2 2 4 5 tem is sent to the impedance controller. Finally, the desired 00 0 1 foot-end force is obtained and then, the joint torque can be calculated by the change of the Jacobian matrix and sent to the robot system. The status of the robot system is also fed The homogeneous transformation matrix of the knee back to the previous process. pitch coordinate system relative to the hip pitch coordinate system is 2.1. One Leg Kinematic Analysis. For realizing the foot trajec- tory control of the leg, the kinematics formula should be 2 3 cosðÞ −θ 0 sinðÞ −θ 0 3 3 derived ﬁrst. To simplify the subsequent analysis of the 6 7 motion control problems, the ﬁrst step is to establish a com- 6 7 01 0 0 HipP 6 7 T = : ð3Þ plete coordinate system for a quadruped robot. Since the 6 7 Knee 6 7 −sinðÞ −θ 0 cosðÞ −θ −L conﬁguration of the four legs of the robot is identical, the 3 3 2 4 5 only diﬀerence is the position of the hip joint relative to 00 0 1 the centre of mass (COM) of the robot. Therefore, the kine- matics analysis of the foot-end with the hip coordinate sys- The homogeneous transformation matrix of the ankle tem is consistent for all four legs. Here, only the right front pitch coordinate system relative to the knee pitch coordinate leg is selected for the analysis in this section. The following system is joint coordinate systems are established according to the Denavit-Hartenberg (D-H) method: the hip rolling coordi- 2 3 nate system ∑ , the hip pitch coordinate system ∑ , HipR HipP cosðÞ θ 0 sinðÞ θ 0 4 4 the knee pitch coordinate system ∑ , the ankle pitch 6 7 Knee 6 7 01 0 0 coordinate system ∑ , and the foot-end coordinate sys- 6 7 Knee Ankle T = , ð4Þ 6 7 Ankle tem ∑ are shown in Figure 2. The blue line in the ﬁgure 6 7 Foot −sinðÞ θ 0 cosðÞ θ −L 4 4 3 4 5 shows the three links of the leg; the red line shows the axes of 00 0 1 the leg rotating joints. The coordinate system of each link, as well as four joint angles, is shown in Figure 3 in detail. The left picture The homogeneous transformation matrix of the foot-end is the front view, where the X axis of the hip torso BH coordinate system relative to the ankle pitch coordinate coordinate system is perpendicular to the paper surface. system is The right picture is a schematic plan view of the right front leg, the rotation axes of the hip pitch, knee pitch, 2 3 100 0 and ankle pitch joints which are oriented perpendicular 6 7 to the paper in. In addition, according to the right-hand 6 7 010 0 6 7 Ankle rule, from the foot-end to the hip joint, the ankle pitch T = : ð5Þ 6 7 Foot 6 7 joint θ , knee pitch joint θ , hip pitch joint θ , and hip 001 −L 4 3 2 4 5 roll joint θ are deﬁned. 000 1 According to the given kinematic model of the quadru- ped robot, from the hip pitch coordinate system ∑ to the BH foot-end coordinate system ∑ , the homogeneous trans- At last, by integrating these transformation matrixes, the Foot formation matrix between the neighbouring link coordinate (6) and (7) can be derived systems can be derived. Firstly, the homogeneous transfor- mation matrix of the hip rolling coordinate system relative HipR HipP BH BH Knee Ankle T = T ⋅ T ⋅ T ⋅ T ⋅ T, ð6Þ Foot HipR HipP Knee Ankle Foot to the hip torso coordinate system is 4 Applied Bionics and Biomechanics 2 3 From equation (7), we get c 0 s −L s − L s − L s 2−3+4 2−3+4 4 2−3+4 3 2−3 2 2 6 7 6 7 −s s c s c −L c −ðÞ L c + L c + L c s 1 2−3+4 1 1 2−3+4 1 1 2 2 3 2−3 4 2−3+4 1 BH 6 7 BH T = 6 7, P + L cos θ Foot sin θ 1 1 y 1 6 7 −c s −s c c L s − L c + L c + L c c ðÞ 1 2−3+4 1 1 2−3+4 1 1 2 2 3 2−3 4 2−3+4 1 = : ð11Þ 4 5 BH cos θ P + L sin θ 1 1 00 0 1 ð7Þ Further, where c represents cos θ , s represents sin θ , c represents 1 1 1 1 2 BH BH cos θ ,c represents cos ðθ − θ Þ,s represents sin ðθ − − P − P 2 2−3 2 3 2−3 2 y qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin θ − qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos θ 1 1 θ Þ,c represents cos ðθ − θ + θ Þ, and s represents 3 2−3+4 2 3 4 2−3+4 2 2 2 2 BH BH BH BH P + P P + P sin ðθ − θ + θ Þ. y z y z 2 3 4 ð12Þ Therefore, according to the positive kinematic analysis, −L = qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : in the case of the known joint angle θ , θ , θ , θ , the position 1 2 3 4 2 2 BH BH P + P and orientation of the foot-end relative to the hip torso coor- y z dinate system can be obtained. For calculating the inverse solution, the solvability should be considered for avoiding So, the no solution or multiple solutions. Usually, solving robot kinematic equations by the inverse operation is a nonlinear BH −L problem, and solving forward kinematic problems is to check 1 y θ = arcsin qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ + arctan : ð13Þ BH whether the target point is in the working space. Therefore, 2 2 BH BH P P + P y z for deciding the existence of the robot inverse kinematics solution, the robot leg’s workspace should be calculated. For ensuring that the inverse kinematic is solvable, the foot From equation (7), we can also get of the quadruped robot must be within the workspace of o o the leg joint. According to the design index θ ∈ ½−20 ,20 BH P = − L + L + L cos θ sin θ + L sin θ cos θ : ðÞ o o o o x 4 2 3 3 2 3 3 2 ,θ ∈ ½−50 ,50 , θ ∈ ½−120 , 120 , a series of coordinate 2 3 ð14Þ points can be obtained with diﬀerent joint angles. A point cloud map indicting the whole working space of the leg is shown in Figures 4 and 5. Since θ has been given by equation (10), so For the solution of inverse kinematic problems, there are mainly two types of closed-form solutions (analytic solu- BH tions) and numerical solutions. In this paper, the closed solu- θ = arcsin qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 tion method is used to solve the analytical solution. Because L + L + L cos θ + L sin θ ðÞ ðÞ 4 2 3 3 3 3 ð15Þ the quadruped robot has the same leg conﬁguration, only L sin θ 3 3 the right front leg is considered to establish its inverse kine- + arctan : L + L + L cos θ matics model. From equation (7), we can get 4 2 3 3 2 2 2 Formula (10), formula (13), and formula (15) are the BH BH BH 2 2 2 2 P + P + P = L + L + L + L +2L L cos θ x y z 1 2 3 4 2 3 3 inverse kinematic equations of the leg, and the inverse +2L L cos θ +2L L cosðÞ θ − θ : solution is the only solution. Thus, if we know the coordi- 3 4 4 2 4 3 4 nates of the foot-end of any leg in the hip joint body coor- ð8Þ dinate system, we can solve the requirement of each joint angle. Because of the parallelogram structure, it is structurally 2.2. Velocity Jacobian Matrix. In this section, the velocity guaranteed that the (8) simpliﬁes to Jacobian matrix is derived which is useful for further leg con- trol such as the transformation of forces and torques from the 2 2 2 BH BH BH 2 2 2 2 P + P + P = L + L + L + L x y z 1 2 3 4 foot-end to the joints. The deﬁnition of the Jacobian matrix is as follows: +2ðÞ L L + L L cos θ +2L L : 2 3 3 4 3 2 4 ð9Þ 2 3 BH BH BH ∂ P ∂ P ∂ P x x x 6 7 ∂θ ∂θ ∂θ 6 1 2 3 7 So, we can obtain 6 7 6 BH BH BH 7 ∂ P ∂ P ∂ P 6 7 y y y J = : ð16Þ ! 6 7 2 2 2 BH BH BH 2 2 2 2 6 7 ∂θ ∂θ ∂θ P + P + P − L − L − L − L − 2L L 1 2 3 4 2 4 1 2 3 x y z 6 7 θ = θ = arccos : 3 4 6 7 2 L L + L L BH BH BH ðÞ 4 5 2 3 3 4 ∂ P ∂ P ∂ P z z z ð10Þ ∂θ ∂θ ∂θ 1 2 3 Applied Bionics and Biomechanics 5 0.3 0.2 0.1 −0.1 −0.2 −0.3 −0.4 −0.5 0.1 0.05 0.5 0.4 0.3 −0.05 0.2 −0.1 0.1 Y(m) −0.15 −0.1 −0.2 X (m) −0.2 −0.3 −0.4 −0.25 −0.5 Figure 4: 3D workspace point map of one leg. 0.3 Table 1: The twelve control points of the Bézier curve. x (mm) y (mm) 0.2 c -170 460 0.1 -280.5 460 0 c -300 361.1 -300 361.1 –0.1 c -300 361.1 –0.2 c 0 361.1 c 0 361.1 –0.3 0 321.4 –0.4 303.2 321.4 303.2 321.4 –0.5 –0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5 282.6 460 X (m) c 170 460 Figure 5: X Z -plane workspace point map. HipR HipR sive loadings will be added on the legs; it will cause a Then, according to the equation of positive kinematics, great rigid impact with only position control. Since the the Jacobian matrix of the velocity from the leg joint coordi- swing phase and the stance phase have diﬀerent dynamic nates to the foot can be obtained as characteristics, the trajectories of the two phases are designed individually. 2 3 Later, for tracking the trajectory, two control methods 0 l c + l c l c 2 2 3 23 3 23 6 7 will be compared which are position control and impedance 6 7 J = l s + l c c + l c c −s l s + l s −l s s : ðÞ 1 1 2 1 2 3 1 23 1 3 23 2 2 3 1 23 control. 4 5 The swing-phase trajectories are designed from a Bézier −l c + l s c + l s c c l s + l s l c s ðÞ 1 1 2 1 2 3 1 23 1 3 23 2 2 3 1 23 curve deﬁned by twelve control points, and stance-phase tra- ð17Þ jectories are designed as part of sinusoidal wave which has a good performance in smoothness and was also used in other 2.3. Foot-End Trajectory Planning. For obtaining better per- robots [17]. formance of the robot, we should design the foot-end tra- jectory properly. During the swing phase, the leg is not 2.3.1. Trajectory Design for the Swing Phase. The swing- aﬀected by contact force, so a higher speed and accelera- phase trajectory design should guarantee enough ground tion can be achieved. During the stance phase, the exten- clearance for avoiding obstacles and reduce energy losses Z (m) Z (m) 6 Applied Bionics and Biomechanics –100 X axis Y axis c7 c8,c9 c2,c3,c4 c5,c6 c0 c10 c1 c11 –300 –200 –100 0 100 200 300 X (mm) Figure 6: The desired trajectory decided by 12 control points of the Bézier curve. which can avoid obstacles in the swing phase. The Bézier curve formula determined by the normalization parameter SW S ðtÞ ∈ ½0, 1 is sw sw sw k sw n−k sw k p t = p S t = 〠 C 1 − S S c , ðÞ ðÞ ðÞ ðÞ ðÞ n k k=0 ð18Þ sw sw sw virtual dp dS dp 1 sw v t = = , sw sw dS dt dS T sw where C represents the number of unordered collection virtual in which k elements are taken from n elements, (n +1) is the number of control points, c is a kth two- dimensional control point where k ∈ f0, ⋯⋯ ,11g. The curve can be generated by twelve control points as shown in Table 1, and the trajectory shape is shown in Figure 6. 2.3.2. Trajectory Design for the Stance Phase. The stance- phase control of each leg will aﬀect the performance of quadruped locomotion via interaction with the ground. Therefore, the planned trajectory should not only consider Figure 7: Virtual impedance model of the leg. the motion requirements but also the interactive force requirements. during touchdown motion [18]. The design of the swing- The stance-phase trajectory is proposed to simply as a phase trajectory should not only approximate the natural sinusoidal wave with two parameters: the half of the stroke behaviour of the leg but also satisfy ground clearance, length L , and the amplitude variable δ. As with the swing span Y (mm) Applied Bionics and Biomechanics 7 Actual trajectory polar 𝜏 , 𝜏 Polar 1 2 Joints of PD controller J (q) polar leg transformation desired Desired trajectory Figure 8: Block diagram for impedance controller. T T phase, the stance-phase trajectory equation is also deter- J = e e = (F – [X X ]𝜃) (F – [X X ]𝜃) G f f G f f st mined by the normalized parameters, S ∈[0,1] st st p t = L 1 − 2S t + P , ðÞ ðÞ x span 0,x 𝜕J π T st st = –2 [X X ] (F – [X X ]𝜃) p t = δ cos p t + P , f f G f f ðÞ ðÞ y x 0,y 𝜕𝜃 2L span 𝜕J st st ð19Þ = 0 2L dp dS span st 𝜕𝜃 v ðÞ t = = − , st dt T dS st st st dp T –1 T dp δπ π y 𝜃 = ([X X ] [X X ]) [X X ] F st x st f f f f f f G v ðÞ t = = sin p ðÞ t : y st x dt T 2L dp st span Figure 9: Derivation block diagram of the least-squares method. Considering that the robot’s leg will touch the ground where x is the position, m is the mass, b is the damping, k is and bear an impact, therefore, an impedance control should the stiﬀness, and f is the force applied by the user. If b or k in be used to resist the impact during the stance phase and will the impedance coeﬃcient is set to be large, it is called high be introduced in the next part. impedance; if b or k is set small, it becomes low impedance. 2.4. Impedance Controller Design. In the previous section, we In this paper, virtual leg impedance is created in the polar discussed the planning for robot’s foot-end trajectory. If the coordinate as shown in Figure 7. robot has no contact with the external environment, pure The control formula can be derived as motion control is enough for trajectory tracking. However, quadruped robots are high dynamic robots that their feet will f − f = mðÞ €x − €x + bðÞ x_ − x_ +kxðÞ − x , ð21Þ control est d d d contact the ground frequently during the movement. In this case, the space constraint brought by the environment will hinder the tracking movement of the robot end eﬀector. where f is the control force sent to the controller; f is control est € _ Therefore, for ensuring the compliance during the move- the estimated ground reaction force; x , x , x are desired d d d ment, the impedance control is used where the leg will be acceleration, velocity, and position; and €x, x_, x are actual acceleration, velocity, and position. Considering that the vir- imitating mass, spring, and damper properties. Based on this, the robot will present virtual mass, stiﬀness, and damping tual mass has no signiﬁcant eﬀect on the impedance eﬀect of characteristics during movement [4]. the robot’s legs, no virtual mass term is added to the control A schematic diagram of a one-dimensional mass- algorithm in this paper. damping-spring model is shown in Figure 7. Impedance is The Jacobian from the hip/shoulder to foot-end in the polar coordinate system is obtained by the transformation. used to describe the behavior of a robot. Diﬀerent impedance parameters can be set to give diﬀerent dynamic characteris- The position relationship between the Cartesian coordinate tics of the robot. system and the polar coordinate system is The dynamics for a one-dof robot rendering an imped- "#"# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ance can be written 2 2 x + y virtual = : ð22Þ m€x + bx_ + kx = f , ð20Þ arctanðÞ x/y virtual 8 Applied Bionics and Biomechanics (a) (b) Figure 10: Experimental platform. (a) One leg experimental platform. (b) Trajectory measurement with the NDI Optotrak Certus sensor. 10 20 Apply external force 5 10 0 0 –5 –10 Apply external counterforce –20 –10 0 2 4 6 8 1012141618 Time (sec) Figure 11: The actual torque and position of the hip joint change after applying force. So the Jacobian from the Cartesian coordinate system to J ðx, yÞJ ðqÞ. And the block diagram for the imped- polar Cartesian the polar coordinate system is ance controller can be shown in Figure 8. 2 3 x y pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2.5. Ground Compliance Estimation. For the above analysis, 6 7 x + y x + y 6 7 J ðÞ x, y = : ð23Þ we assumed that the ground is completely rigid, but for a real 6 7 polar 4 1 −x 5 application, the assumption is limited. For example, if the 2 2 2 2 2 yx /y +1 y x /y +1 ðÞ ðÞ robot is moving on a concrete ﬂoor or an asphalt road, we can think that the assumption is completely valid. But when Then, impedance control law can be derived as the robot moves in an environment like grass, marshes, and snow, there will be a big diﬀerence. Therefore, the identiﬁca- "# "# τ k e + b e_ tion of ground compliance is important. Through the system 1 ρ ρ ρ ρ = J ðÞ q , ð24Þ polar identiﬁcation method, we can get the stiﬀness and damping τ k e + b e_ 2 θ θ θ θ characteristics of the contact ground, and then, further oper- ations to achieve the corresponding impedance characteris- _ _ where e , e , e , and e are radial position error, radial veloc- ρ ρ θ θ tics can be implemented. ity error, angular position error, and angular velocity error Among the system identiﬁcation theory, the least-squares between the actual trajectory with designed trajectory, method is widely used, and the eﬀect is also excellent. There- respectively. J ðqÞ is the Jacobian from hip/shoulder to fore, we use the least-squares method to estimate the ground polar compliance parameters. foot-end in the polar coordinate system, and J ðqÞ = polar Actual torque (N m) Actual position (deg) Applied Bionics and Biomechanics 9 –2 –4 –6 –10 –5 0 5 10 Actual position (deg) Actual torque vs. Actual position linear fitting Figure 12: The actual torque change with the position of the hip joint and the ﬁtting straight line. a good parameter estimate can be obtained and the data sat- uration phenomenon can be eﬀectively improved. To implement the estimation of parameters, we construct the following dataset F =½ f ðÞ n f ðÞ n − 1 ⋯ f ðÞ n − k , G G G G X = x ðÞ n xðÞ n − 1 ⋯ xðÞ n − k , ð26Þ f f f f X = x_ ðÞ n x_ðÞ n − 1 ⋯ x_ðÞ n − k : f f f f By estimating θ and F as the optimal estimate of θ = ½k b and F , we can get G G G hi ̂ _ F = X X θ : ð27Þ f f Figure 13: Diagram of free dropping apparatus of the single-leg Then, by deﬁning e = F − F and J = e e, and from that, G G prototype. the least-squares estimation requires that the sum of the squares of the residuals be the smallest; we can obtain the parameters as shown in Figure 9. Taking into account the storage performance and com- Moreover, considering that recent samples make more putational performance of industrial controllers, we estimate great impact on the results, we use a weighting matrix Q ∈ the ground parameters using the least squares method in lim- k×k ℝ to penalize the error on most recent sample compared ited memory, which limit the estimated range. The ground to the less recent ones and thus, giving more importance to reaction force (GRF) is described as the most recent samples. And the result is f = k x + b x , ð25Þ G f G f hi hi −1hi T T _ _ _ θ = X X Q X X X X F : f f f f f f G where k and b are ground stiﬀness and damping coeﬃ- G G cients; that is, the values we need to identify by the system ð28Þ identiﬁcation x and f are the depth and the ground reac- tion force. Then, we can estimate the ground parameters by T According to the estimates θ = , we can ½ k∧ b∧ G G the least squares method in limited memory. acquire At every time instant n, we gather samples from the pre- vious k time instances and compute the ground parameters. ̂ ̂ ̂ f = k x + b x_ : ð29Þ G G f G f By choosing the appropriate k value, i.e., the deﬁned range, Actual torque (N·m) 10 Applied Bionics and Biomechanics Current variation –5 Free drop Stop under external force –10 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Position variation 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Figure 14: The current and position variation of the knee motor under pure position control. Table 2: Comparison between pure position control and impedance control. Control mode Stability Position steady-state error (deg) Peak current (A) Compliance Impedance control Stable 11.4 4.309 Soft Position control Unstable — 11.03 Hard Current variation Free drop –2 –4 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Position variation 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Figure 15: The current and position variation of the knee motor under impedance control. Then, in the stance phase, for getting better impedance 3. Experimental Results characteristics for the robot, we apply this force to the imped- ance control formula (21), and we can get 3.1. Foot-End Trajectory Experiment. For ensuring that the robot can well perform the planned motion characteristics, an experiment was carried out to verify the robot’s perfor- f − f = m €x − €x + b x_ − x_ +kx − x : ð30Þ ðÞ ðÞ ðÞ control G d d d mance of foot-end trajectory tracking. The experimental platform was constructed as shown in By the least-squares estimation method with limited mem- Figure 10. The NDI Optotrak Certus was used to trace the ory and fading memory for ground parameters, we reasonably presticked marker on the endpoint of the leg, and the actual introduced the estimated value of GRF. Applying the esti- trajectories can be obtained. The accuracy that NDI Optotrak mated GRF to the f impedance controller, the value will Certus can achieve is 0.01 mm and its sampling frequency is desired change with diﬀerent ground conditions, allowing the robot to 100 Hz, which is suﬃcient for our trajectory following the adapt to diﬀerent ground conditions with suitable compliance. acquisition. Position (deg) Current (A) Current (A) Position (deg) Applied Bionics and Biomechanics 11 Because the data collected by the NDI Optotrak Certus is three-dimensional coordinates of a point on its body, and the trajectory points we plan are the two-dimensional coordi- nates with the axis of the hip joint motor as the origin; it is necessary to compare the trajectories after the coordinate transformation. The ﬁrst step is to ﬁnd the homogeneous transformation matrix T from the NDI Optotrak Certus coordinate sys- tem A to the robot coordinate system O. In the coordinate system O, four scattered coordinate points were selected that the foot-end of the robot can reach. Suppose the leg motion is strictly in a plane, and z =0, then Materials with assume that the coordinates of the selected four points are different stiffness O O O O O O O O O O ð x , y , z Þ, ð x , y , z Þ, ð x , y , z Þ, and ð x , 1 1 1 2 2 2 3 3 3 4 O O y , z Þ. 4 4 Simultaneously, the four points are collected by the NDI Optotrak Certus, and the coordinates under frame A are A A A A A A A A A A A ð x , y , z Þ, ð x , y , z Þ, ð x , y , z Þ,and ð x , y , 1 1 1 2 2 2 3 3 3 4 4 Force sensor z Þ. So 2 3 2 3 O A x x 1 1 6 7 6 7 Figure 16: Diagram of the identiﬁcation of various ground O A 6 7 6 7 y y 6 1 7 6 1 7 O materials. = T : ð31Þ 6 7 6 7 O A 6 7 6 7 z z 4 1 5 4 1 5 And the ﬁtting equation is 1 1 y = −0:501x − 0:0003333: ð33Þ After making the same transformation of all four coordi- nates and transforming the ﬁnal matrix equation appropri- 3.3. Comparison between Pure Position Control and ately, we can get Impedance Control. As we know, the essence of impedance control is to control the dynamic relationship between force 2 3 2 3 and position. The pure position control is mainly to achieve −1 O O O O A A A A x x x x x x x x 1 2 3 4 1 2 3 4 an accurate position and does not consider the interaction 6 7 6 7 O O O O A A A A 6 7 6 7 with the outside world. To better understand impedance con- y y y y y y y y O 6 1 2 3 4 7 6 1 2 3 4 7 T = · : 6 7 6 7 trol and pure position control, we have done experiments to O O O O A A A A 6 7 6 7 z z z z z z z z 4 1 2 3 4 5 4 1 2 3 4 5 compare the characteristics of the two control methods. According to the characteristics of the two control methods, 1 111 111 1 we can determine which way is more suitable in the locomo- ð32Þ tion of the quadruped robot. In the previous analysis, we have also seen that in the swing phase, since the foot-end point of the single-leg proto- After ﬁnding the homogeneous transformation matrix type has no contact with the ground, the eﬀects of impedance T , the points collected by the NDI Optotrak Certus are control and pure position control are similar. But in the homogeneously transformed to obtain the actual trajectory stance phase, the foot-end point of the single-leg prototype in the O coordinate system and compared with the planned contacts the ground, and there will be a large impact at the trajectory. moment of contact. If the control strategy does not have a 3.2. Impedance Control Experiment. For the experiment, the certain cushioning eﬀect, it will be harmful to the motor. Therefore, it is necessary to test and analyse the control eﬀect impedance control is based on the control in the torque mode. Therefore, the controller should ﬁrst be switched to of pure position control and impedance control. the torque mode at ﬁrst. In our study, the controller As shown in Figure 13, the experimental procedure is to (C6015, Beckhoﬀ) was chosen with EtherCAT connection set a ﬁxed desired position through the impedance control to each motor. and pure position control in the single-leg prototype in the We observed the relationship between the real-time posi- suspended state (here, the desired force in the impedance tion, the real-time torque, and time by applying external tor- control is set to 0). Then, at a certain height, the single-leg que to the hip joint, as shown in Figure 11. Then, we ﬁt the prototype was dropped freely, and the current and position torque position and get the result as shown in Figure 12. change of the single-leg prototype motor were observed. 12 Applied Bionics and Biomechanics K1 = 110.77 0 10 20 30 40 50 60 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 4 Number of data points Penetration variation (mm) Experimental data points Straight line fitted by least square method (a) K2 = 85.98 0 1020304050 60 0 0.1 0.2 0.3 0.4 Number of data points Penetration variation (mm) Experimental data points Straight line fitted by least square method (b) 2.5 1.5 K3 = 0.69 0.5 0 50 100 150 200 01234 Number of data points Penetration variation (mm) Experimental data points Straight line fitted by least square method (c) Figure 17: Continued. Estimated stiffness (N/mm) Estimated stiffness (N/mm) Estimated stiffness (N/mm) Ground reaction force variation (N) Ground reaction force variation (N) Ground reaction force variation (N) Applied Bionics and Biomechanics 13 0.8 0.6 0.4 K4 = 1.35 0.2 –5 0 1020304050 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Number of data points Penetration variation (mm) Experimental data points Straight line fitted by least square method (d) 1.5 –5 K5 = 1.94 –10 0.5 –15 –20 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 Number of data points Penetration variation (mm) Experimental data points Straight line fitted by least square method (e) Figure 17: The recursive change process of stiﬀness and the ﬁtting straight line. (a) Medium hard rubber. (b) High durometer rubber. (c) Low hardness sponge. (d) Medium hardness sponge. (e) High hardness sponge. It should be noted that the drop height under the imped- to the motor and the outside world, it is necessary to add ance control is 44 mm. Therefore, in the pure position con- external force to force the single-leg prototype to stop beating. trol, the drop height was also set to 44 mm at ﬁrst, but it is impossible to collect appropriate experimental data with high In the impedance control as shown in Figure 15, we can impact. At last, the drop height was determined to be 10 mm. see that although the drop height is 44 mm, which is 4.4 times In addition, through the data measurement and analysis of in the above example, the peak current during the whole drop the two joint motors, the variation of current and position process is only 4.309 A as shown in Table 2, which eﬀectively solves the impact on the motor during the dropping process. values of the hip motor during the falling process is not par- ticularly obvious. Therefore, we did not collect the data on hip joint motor, but only current and position acquisition 3.4. Ground Compliance Identiﬁcation. Through the previous of the knee motor. analysis, we set up the corresponding experiments to identify By analyzing the experimental data in Figure 14, we can the stiﬀness of the ground of various materials. know that in the pure position control, even if it drops from A portion of the experimental setup is similar to the pre- a height of 10 mm, the position and current value of the vious motion control experiment where the motion capture motor show a tendency to disperse, that is, an unstable state. system is used to capture the depth of the foot into the mate- And the peak current during the whole process is 11.03 A as rial. Since the stiﬀness is deﬁned as the force acting on the shown in Table 2. In the ﬁnal stage, to avoid serious damage unit displacement, it is also necessary to place a force sensor Estimated stiffness (N/mm) Estimated stiffness (N/mm) Ground reaction force variation (N) Ground reaction force variation (N) 14 Applied Bionics and Biomechanics as shown in Figure 16 at the foot-end to measure the ground State Key Laboratory of Robotics and System (HIT) reaction force when the foot-end is in contact with the (SKLRS201801A02), and the Innovative Research Groups ground. of the National Natural Science Foundation of China The experimental procedure consists of designing a con- (51521003). trol program that allows the foot-end point of the single-leg prototype to move vertically downward from above the ground material and penetrate the material, and the NDI References Optotrak Certus and the force sensor collect the displace- [1] J. Hwangbo, J. Lee, A. 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Applied Bionics and Biomechanics – Hindawi Publishing Corporation
Published: Sep 22, 2020
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