，

（Institute of Automation，Chinese Academy of Sciences，Beijing 100190，China）

Stable Transition of Quadruped Rhythmic Motion Using the Tracking Differentiator

Xiaoqi Li?，Wei Wang and Jianqiang Yi

（Institute of Automation，Chinese Academy of Sciences，Beijing 100190，China）

Since the quadruped robot possesses predominant environmental adaptability，it is expected to be employed in nature environments.In some situations，such as ice surface and tight space，the quadruped robot is required to lower the height of center of gravity（COG）to enhance the stability and maneuverability.To properly handle these situations，a quadruped controller based on the central pattern generator（CPG）model，the discrete tracking differentiator（TD）and proportional-derivative（PD）sub-controllers is presented.The CPG is used to generate basic rhythmic motion for the quadruped robot.The discrete TD is not only creatively employed to implement the transition between two different rhythmic medium values of the CPG which results in the adjustment of the height of COG of the quadruped robot，but also modified to control the transition duration which enables the quadruped robot to achieve the stable transition.Additionally，two specific PD sub-controllers are constructed to adjust the oscillation amplitude of the CPG，so as to avoid the severe deviation in the transverse direction during transition locomotion.Finally，the controller is validated on a quadruped model.A tunnel with variable height is built for the quadruped model to travel through.The simulation demonstrates the severe deviation without the PD sub-controllers，and the reduced deviation with the PD sub-controllers.

quadruped robot；center of gravity；central pattern generator；discrete tracking differentiator；proportional-derivative sub-controller

1 Introduction

Quadruped robots have a promising perspective because of their significant stability of locomotion and powerful maneuvering capabilities on rough terrains.A great number of quadruped robots have been developed，such as Bobcat-robot［1-2］，HyQ［3-4］，Cheetah-cub［5］，Tekken［6］，RoboCat-1［7］，MIT Cheetah［8］，PUPPY II［9］，LittleDog［10］and BigDog［11］.The majority of these quadruped robots have been tested for maneuvering capability which is a core task for quadruped robots.Compared with a fixed spine，active spine supported actuation leads to faster locomotion，less foot sliding and a higher stability for the Bobcat-robot running in bound gaits［1］.In RoboCat-1，active compliance control via force feedback，and angular momentum control via gyro sensing，are synthesized to realize cyclic，actively compliant and dynamically-balanced jumping and trotting on quadruped locomotion over rough terrains［7］.The design principles for highly efficient legged robots are implemented in the MIT Cheetah robot［8］.BigDog［11］with rough-terrain mobility is able to locomote on real world rough terrains.

To improve the robustness，researchers have proposed a large number of approaches.One approach is the CPG model which is based on biological studies to generate rhythmic motion.There has been already much research devoted to the CPG model.A control strategy which uses the CPG model，stumbling correction，leg extension reflexes，and model-based posture control mechanisms，is utilized to realize dynamic locomotion over unperceived uneven terrains，steps，and slopes［12］.A bio-inspired architecture，based on the CPG and a tonic signal，is proposed to easily select and switch between different gaits to improve the robot stability and smoothness while locomoting［13］.A control architecture，proposed with a 3-D workspace trajectory generator based on the CPG and a motion engine，is able to genenrate adaptive workspace trajectories online by tuning the parameters of the CPG network to adapt to various terrains［14］.A combination of discrete and rhythmic motor primitives is proposed and applied to interactive drumming and infant crawling in a humanoid robot［15］.A gait multi-objective optimization system that combines the CPG model and a multi-objective evolutionary algorithm is proposed to perform the required walking gait［16］.

In certain circumstances，such as ice surface and tight space，it is necessary for the quadruped robot to lower the height of COG to improve the flexibility and maneuverability.The COG trajectory is of great importance to stability and robustness in locomotion.A novel parameterization of the COG trajectory which guarantees continuous velocity and acceleration profiles of the leg movement has been implemented on Little-Dog to traverse complex rocky and sloped terrains［17］. An adjustment approach of the COG［18］is designed to realize stable quadruped walking.The COG trajectory is established by online autonomous learning under the objective of locomotion stability and naturalness，and implemented on a real humanoid robot PKU-HR4［19］. A hierarchical control architecture，presented and experimented on LittleDog to walk over rough terrain，plans a set of footprints across the terrain，trajectories for the robot feet and the COG，and tracks these desired trajectories［20］.

The adjustment of the height of COG is implemented by changing rhythmic medium value，so a challenging issue is presented，that is，how to transit the motion between two different rhythmic medium values.Wu et al.presented a stable quadruped walking approach with the COG modulation to pass through tight space with free-collision，however，there exists large deviation in transverse direction［18］.

In this contribution，a controller based on the CPG，discrete TD and PD sub-controllers is presented to control the quadruped motion transition.As usual，we utilize the CPG to generate basic rhythmic locomotion.The discrete TD is usually used for time delay，multivariable decoupling control，cascade control and parallel system control［21］.However，we argue that the discrete TD can be employed to realize smooth transition of the quadruped rhythmic motion，and so we develop the discrete TD whose transition duration can be controlled in an effort to transit the quadruped motion between two different rhythmic medium values smoothly.Two PD sub-controllers based on position and orientation are implemented to adjust the oscillation amplitude of the CPG，such that the severe deviation in the transverse direction is reduced significantly.

Section 2 depicts the Hopf oscillators and the modified discrete TD.The PD sub-controllers based on position and orientation are introduced in Section 3. Section 4 depicts the quadruped robot model and the control parameters.Section 5 gives the simulation and discusses the simulation results.Finally，Section 6 concludes the work and gives an outlook on the future work.

2 Hopf Oscillators and Modified Discrete Tracking Differentiator

The controller is composed of three modules，as shown in Fig.1：the CPG to generate rhythmic motion，the modified discrete TD，and two PD sub-controllers based on position and orientation.The CPG network is constructed using coupled Hopf nonlinear oscillators to generate trajectories for hip pitch joints and knee pitch joints.are rhythmic signals generated by the CPG and implemented to the quadruped model，respectively.denote the amplitude ofandrespectively.The modified discrete TD with outputis utilized to track the changed rhythmic medium value smoothly.The PD sub-controller-I with feedback（the yaw angle of the quadruped robot），and PD sub-controller-II with feedback（the position component of footprints in the transverse direction），are cooperated to modify the amplitude of the CPG to eliminate the deviation in the transverse direction during locomotion.are the modulation factors generated by the PD sub-controller-I and the PD subcontroller-II，respectively.

Fig.1 The structure of the controller

2.1 Rhythmic Motion Based on Hopf Oscillators

The CPG，inspired from biological concepts，is capable of autonomously producing coordinated rhythmic output signals for locomotion in a robust and flexible way.The Hopf oscillator is one model of the CPG and is adapted to generate the rhythmic signals for movement.The Hopf oscillator has many advantages，high stability against perturbation，strong phase lockingfeature，smooth trajectories modulated by a simple parameter change，etc.The Hopf oscillator is defined by the following nonlinear differential equations［22］：

In order to independently control the frequencies of the ascending and descending phases of the oscillation，we use the following equation［22］，

whereω（rad／s），determined byωswingandωstance，is the frequency of the oscillation.ωswingandωstancecontrol the durations of the ascending and descending phases ofxtrajectory，respectively.

We utilize a Hopf oscillator for each leg of the quadruped robot.As shown in Fig.2，the CPG network is constructed by connecting the Hopf oscillator of each leg.The coupled Hopf oscillators oscillate in the same period with a fixed phase difference，which results in the gait，such as walk gait，trot gait，bound gait and gallop gait，given as follows［22］：

wherei，j＝1，2，3，4 correspond to LF，RF，LH and RH，respectively.The abbreviations LF，LH，RF and RH represent left fore，left hind，right fore，and right hind，respectively.kij，specified by the coupling matrices，is the coupling coefficient and determines the type of gait.The other parameters in Eqs.（4）-（5）are defined as above.

Fig.2 The CPG network applied to gait generation

The CPG network generates movement for the hip pitch joints and the knee pitch joints.The four hip joint angles are controlled byxi（i＝1，2，3，4），respectively.The four knee joint angles are controlled as simply as possible.In swing phase，the trajectories for the knee joints areyi（i＝1，2，3，4）and in stance phase，the knee joints do not move.

2.2 Relationship between Joint Angles and the Height of COG of the Quadruped Robot

The quadruped robot benefits from lowering the height of COG to keep stability or increase the maneuverability when it is on ice surface or in some unstructured environments such as tight space.The height of COG of the quadruped robot is changed by adjusting the rhythmic medium values of the hip and knee pitch joints of the four legs，which results in the change of the virtual leg lengths.The virtual leg length is defined as the vertical distance between the hip joint and the foot of the same leg.

When the quadruped robot is in walk gait，there are always three legs in stance phase and one leg in swing phase.It is assumed that the virtual lengths of the two fore legs are same，and the virtual lengths of the two hind legs are same.

As shown in Fig.3，the legs in stance phase are utilized to compute the height of COG.The knee anglesα1andα2are fixed values in stance phase，equal to the rhythmic medium values of the fore knee joint angle and the hind knee joint angle，respectively.The height of COG of the quadruped robot can be computed as follows：

wherem1＝α1-θ1，m2＝α2-θ2．θ1andθ2represent the fore hip joint angle and the hind hip joint angle，respectively.α1andα2are the rhythmic medium values of the fore knee joint angle and the hind knee joint angle，respectively.HCOGrepresents the height of COG of the quadruped robot.θ1，θ2，α1，α2，L1andL2are shown in Fig.3.L1andL2are the length of the thigh and the shank，respectively.

Fig.3 Sketch of the quadruped robot

Eq.（6）shows that the height of COG depends onθ1，θ2，α1andα2．Therefore，to get a smooth COG trajectory，it is crucial to realize smooth transition between two different rhythmic medium values of the hip joint angles and the knee joint angles.

2.3 Modified Discrete Tracking Differentiator

The quadruped robot is able to traverse tight space and enhance stability in certain environment，forexample，ice surface，with the adjustment of the height of COG.Smooth transition between two different rhythmic medium values is of vital importance to realize adjustable height of the quadruped robot’s COG.The discrete tracking differentiator（TD），proved to be convergent［23］，can track the input signal as soon as possible，and generate approximate differential signal. The discrete TD is used to arrange the transient process，filter out the noises in the signal，etc.In this paper，we focus on utilizing the discrete TD to realize smooth transition between two different rhythmic medium values.The discrete TD is shown as follows［21，24］：

wherev1is the output andv2is its derivative.vandhare the input and the step size，respectively.rdetermines the speed of the transition which becomes faster as the parameterrincreases.Further，fh（v1，v2，r，h）is defined as［21，24］：

The transition duration can be controlled by modifying the parameterr．However，the transition duration in traditional discrete TD is implicit.We hope to control the transient duration directly.Assuming the transition duration isT，the step response curve generated by Eqs.（7）-（9）has the following characteristics［25］：

1）In the interval（0，T／2），the step response curve rises with acceleration equal to parameterrand the velocity increases from 0 tor．

2）In the interval（T／2，T），the step response curve rises with acceleration equal to the opposite number of parameterrand the velocity decreases fromrto 0.

Sothe following function is defined：

whereTcontrols the transient duration.vcurandvlastdenote the present and last medium set values，respectively.The outputv1can track the set valuevarbitrarily fast as long asTis chosen arbitrarily small without any overshoot.

Here letvtake the rhythmic medium value.The outputv1realizes smooth transition between two different rhythmic medium values.For example，the rhythmic medium valuevstarts from rest with 0.0，switches to-0.52 after 3.5 s，and to 0.0 after 20.0 s. The outputsv1depending on the different value ofTare shown in Fig.4.WhenTis small，such asT＝1．0，v1tracks the set value fast.WithTincreases，the transient profile becomes gentler.WhenTis large enough，for example，T＝12.0，the transient profile is the gentlest of the three outputs.It is obvious that the transient profile becomes gentler with the increase of the parameterT，which means that the transition between two different rhythmic medium values is stable. However，the transient time is required to be as small as possible considering the promptness.Therefore we should manage tradeoffs between high stability and promptness so as to choose appropriate value ofT．

Fig.4 The outputsv1with different values of parameterT

3 PD Sub-controllers for Correction of the COG Deviation

The quadruped robot deviates in the transverse direction during the rhythmic medium value transition. The mediolateral reaction force，one component of ground reaction force，is the factor contributing to the deviation［26］.

Inspired by the turning maneuver technique for a quadruped robot with only the hip pitch and the knee pitch for each leg［26］，we design two PD subcontrollers based on position and orientation that are capable of responding immediately to the deviation in the transverse direction by modifying the amplitude of the CPG to eliminate the deviation.The positions of the quadruped robot’s footprints and the yaw angle of the quadruped robot can be measured by the high-precision global position system（GPS）and the angular transducer，respectively.To analyze if there is deviation，the follows are defined：

wherei＝LF，LH，RF，RH.denote the current and desired position components of footprints in the transverse direction，respectively.θyawandθdesare the current and desired yaw angles of the quadrupedrobot，respectively.is the error between the current and desired position components of footprints in the transverse direction.is the error between the current and desired yaw angle.The quadruped robot deviates ifis not zero.According to the sign（positive or negative）ofthe deviation direction can be acquired.

The PD sub-controllers are designed as：

Finally，the structure of the designed PD subcontrollers，described by Eqs.（11）-（15），is shown in Fig.5.

After a detailed description of the controller which consists of the CPG，the modified discrete TD，and the PD sub-controllers，the detailed control architecture is depicted in Fig.6.

Fig.5 The structure of the PD sub-controllers to modify the amplitude of rhythmic signals

Fig.6 The detailed control architecture of the controller

4 Quadruped Robot Model and Control Parameters

4.1 Quadruped Robot Model

The simulation is performed in MSC.ADAMS simulation platform with a simulated quadruped robot model.The quadruped model in Fig.7 is 0.63 m in height and 50 kg in weight.Each leg has three DOFs，including hip yaw，hip pitch，and knee pitch.The hip yaw joint of each leg is set to fixed in the simulation. The initial values of the hip joint angleθ0and the knee joint angleα0are 30°and 60°，respectively.

4.2 Control Parameters

We implement the controller in MATLAB／Simulink for the quadruped robot model，and use dynamic-link library in MATLAB／Simulink to control the quadruped robot model in ADAMS.There are three kinds of control parameters to be set，including the CPG parameters，discrete TD parameters and PD subcontroller gains.

Fig.7 The quadruped model in MSC.ADAMS simulation platform

CPG parameters：the quadruped robot model is implemented and simulated in walk gait，so the coupled Hopf oscillators’phase difference in turn is π／4．The coupling coefficientkijis set to-1 when the oscillators are inhibitive effect，1 otherwize.The state variablexiis used as the rhythmic signal for the hip joint of the quadruped model.Each knee joint is controlled by the state variableyiduring swing phase，and does not move during stance phase.

Discrete TD parameters：four stages about rhythmic medium values are set.As shown in pink，dotted line in Fig.8，the rhythmic medium values of the LH and RH hip angles in Stages I，II，III and IV are set to 0，π／8，π／6 and 0，respectively.The rhythmic medium values of the LF and RF hip joints angles（in blue，dash line in Fig.8）in Stages I，II，III and IV are set to 0，-π／8，-π／6 and 0，respectively.The rhythmic medium values of the LF and RF knee joints angles（in black，plain line in Fig.8）in Stages I，II，III and IV are set to 0，π／4，π／3 and 0，respectively. The rhythmic medium values of the LH and RH knee angles（in red，dash-dotted line in Fig.8）in Stages I，II，III and IV are set to 0，-π／4，-π／3 and 0，respectively.

PD sub-controller gains：the derivative gains of the two PD sub-controllers are firstly set to zero to adjust the proportional gains.Kp1andKp2gradually increase to appropriate values which lead to an deviation curve with quick response and small overshoot.Then the derivative gainsTd1andTd2are increased from zero to appropriate values to remove the static errors and enhance the stability.Kp1，Kp2，Td1andTd2are finally set to 3／2，3／2，1／2 and 1／2，respectively.

Finally，the control signals ofthe hip joints and the knee joints are shown in Figs.9（a）and 9（b），respectively.In Fig.9（a），the control signals of the LF，LH，RF and RH hip joints are in black dashdotted line，blue plain line，red dotted line and pink dash line，respectively.In Fig.9（b），the control signals of the LF，LH，RF and RH knee joints are in black plain line，red dotted line，pink dash-dotted line and blue dash line，respectively.The quadruped model walks at an average speed of 0.23 m／s.

Fig.8 Rhythmic medium values of the hip pitch joints and the knee pitch joints

Fig.9 Control signals implemented in the quadruped model

5 Simulation Results and Discussions

In order to demonstrate the effect of the controller，a tunnel with variable height is constructed for the quadruped model to travel through.The first segment of the tunnel is 550 mm in height，while the second segment of the tunnel is 450 mm in height，as shown in Fig.10. The simulation is implemented in a time window of 30 s.

As shown in Fig.10，the COG trajectory isrecorded by the gray line in side view.Obviously，the COG trajectory is well adaptive to variable height of the tunnel，and the simulated robot can travel through the tunnel successfully.

Fig.10 Side view of simulation scenarios

Fig.11 depicts snapshots of the simulated model traversing through the tunnel with variable height by adjusting the rhythmic medium values of the CPG smoothly and flexibly.As shown，the quadruped robot smoothly adjusts the virtual leg lengths in order to travel through the first segment of the tunnel at 6.2 s ahead.The virtual leg lengths are further reduced at 10.9 s to travel through the relatively lower segment of the tunnel.The height of COG of the quadruped model restores to its origin when it is about to walk out of the tunnel.

Fig.11 Snapshots of the quadruped robot model smoothly traversing the height-changed tunnel

As shown in Fig.12，the COG trajectory in thex-zplane，i.e.the sagittal plane，has three transitions within 6.76 meters displacement inx-axis.The height of COG changes from 650 mm to 475 mm in the first transition and then reduces to 400 mm in the second transition，then restores to 650 mm in the third transition.All transitions are smooth so that the simulated robot can travel through the tunnel stably.

Fig.12 The COG trajectory in the sagittal plane

The modified discrete TD is employed to implement the transition between two different rhythmic medium values of the CPG.But，the severe deviation occurs in the transverse direction.So we add two PD sub-controllers based on position and orientation to the controller to reduce the deviation.To show the effect of the two PD sub-controllers，the COG deviations without the PD sub-controllers and with the PD sub-controllers are compared.Fig.13 demonstrates the COG trajectories in thex-yplane，i.e.the ground plane，and the left fore and right fore footprints from the simulation in Fig.10.In a 6.76-meter locomotion test，the COG deviation ranges from-25 mm to 175 mm without the PD sub-controllers（Fig.13（a）），and swings appropriately within-20 mm and 20 mm with the PD sub-controllers（Fig.13（b））.It is shown that the deviation in the transverse direction can be controlled effectively with the PD sub-controllers.

Fig.13 The COG trajectories and the LF and RF footprints（top view）are illustrated in thex-yplane，i.e.the ground plane

6 Conclusions and Future Work

The discrete TD is used to implement the stable transition between two different rhythmic medium values for the quadruped locomotion.In our presented controller，the discrete TD is employed to control the transition.Two PD sub-controllers are designed to adjust the oscillation amplitude of the CPG model，such that the severe deviation in the transverse direction can be reduced effectively.Again，the discrete TD is modified，so it is available to control the transition duration.

The controller is carried out on a simulated quadruped robot.To show the effect of the controller，we build a height-changed tunnel for the simulatedquadruped robot to traverse.The results indicate the controller can generate smooth rhythmic medium value transition with the reduced deviation in the transverse direction.

Future work will address adding modules to enable the quadruped robot to traverse more confined environments.Furthermore，we will evaluate the controller on the real quadruped robot，and improve the presented controller according to the experiment results.

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TP242.3

：

：1005-9113（2015）05-0009-08

10.11916／j.issn.1005-9113.2015.05.002

2014-07-22.

Sponsored by the National Natural Science Foundation of China（Grant No.61375101）.

?Corresponding author.E-mail：xiaoqi.li＠ia.ac.cn.